estimating the critical height of unsupported trenches in

Estimating the Critical Height of Unsupported Trenches in Unsaturated Soil

by

Adin Richard

B.Sc.E. (Civil Engineering), University of New Brunswick, 2016

A Thesis Submitted in Partial Fulfillment of the requirements for the Degree of

Master of Science in Engineering

In the Graduate Academic Unit of Civil Engineering

Supervisor: Won Taek Oh, Ph.D., P.Eng., Department of Civil Engineering

Examining Board: Kaveh Arjomandi, Ph.D., P.Eng., Department of Civil Engineering Othman Nasir, Ph.D., P.Eng., Department of Civil Engineering Mohsen Mohammadi, Ph.D., Department of Mechanical Engineering

This Thesis is accepted by the Dean of Graduate Studies

THE UNIVERSITY OF NEW BRUNSWICK

August 2018

© Adin A. Richard, 2018

ii

ABSTRACT

Trenching (or excavating) is integral to construction activities in the practice of

geotechnical, mining, tunnelling, and geo-environmental engineering. Many projects are

initiated by excavating trenches for infrastructure to be installed. Workers are often

required to enter trenches during the construction process, which may present serious risks.

Trench failures can result in death and/or damage to adjacent properties, therefore trenches

should be excavated with extreme precaution. The critical height (i.e. maximum depth that

can be excavated without failure) is the most important design consideration for ensuring

the stability of unsupported trenches. Excavation work is often done in unsaturated soils,

in which case the critical height of unsupported trenches can be more reliably estimated by

considering the influence of matric suction.

The purpose of this study is to investigate the influence of soil type, the matric suction

distribution, and surcharge pressure in assessing the critical height of unsupported trenches.

Extended Rankine earth pressure theory, the limit equilibrium method, and finite element

analyses were used to estimate the critical height of unsupported trenches in two

unsaturated soils.

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ACKNOWLEDGEMENTS

Thank you to all friends, family, and faculty for the emotional and the financial support.

There are countless individuals that made this degree possible. First of all, my parents,

Sean and Holly, you have positioned me to succeed and for that I will be forever grateful.

I am especially thankful for the number of times you have been my shuttle for kayaking.

Dad, I am sorry for the number of times us dirtbags have piled in the truck and messed it

up. Mom, thank you for continually making him take us. I genuinely could not have asked

for better parents. Stuart, my brother, I am most thankful for the number of times you

cooked me supper over the last two years. Seriously, you should open a stir-fry restaurant

(name it Stu-Fry ©). I was fortunate enough to share an office with two great colleagues,

Vitus Ileme, and Greg Brennan. I am happy to call both of you my friends. Our creative

discussions always kept me thinking. Vitus, your positive attitude and composure always

helped me on my down days. Greg, your knack for problem solving is something to admire.

Thank you both for all that you contributed to this thesis. Dr. Haralampides and Dr.

MacQuarrie, I thoroughly enjoyed each class I had with you going all the way back to

undergrad. Thank you both for helping round out my knowledge in hydraulics and

hydrology. Lastly, thank you, my supervisor, Dr. Oh. I am honoured to have been accepted

as your first ever graduate student. I think we both took a leap of faith. I did not actually

understand soil mechanics until that first semester when you taught us the ‘Mechanics of

Unsaturated Soils’. You have brought me a long way. And of course, thank you to my

friends that were always there to help me unwind. I could not have done without every one

of you!

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TABLE OF CONTENTS

ABSTRACT ................................................................................................................. ii

ACKNOWLEDGEMENTS ............................................................................................ iii

TABLE OF CONTENTS ................................................................................................ iv

LIST OF TABLES ........................................................................................................... vi

LIST OF FIGURES ........................................................................................................ vii

LIST OF SYMBOLS & ABBREVIATIONS ............................................................... xii

CHAPTER 1 - Introduction ............................................................................................. 1

1.1 Problem Statement ............................................................................................... 1

1.2 Objectives of the Thesis ....................................................................................... 2

1.3 Scope of the Thesis .............................................................................................. 3

1.4 Organization of the Thesis ................................................................................... 3

CHAPTER 2 – Trenching Hazards and Regulations in Canada ................................. 5

2.1 Trench Collapse Statistics .................................................................................... 6

2.2 Trench Failure Mechanisms ................................................................................. 8

2.3 Trench Failure Modes ........................................................................................ 10

2.4 Work Protection Methods .................................................................................. 12

2.5 Trench Economics .............................................................................................. 16

CHAPTER 3 – Theoretical Background ...................................................................... 18

3.1 Properties of Unsaturated Soil ............................................................................ 18

3.2 Shear Strength of Unsaturated Soil .................................................................... 23

3.3 Critical Height of Unsupported Vertical Trenches ............................................ 29

3.3.1 Rankine’s Earth Pressure Theory (1857) .................................................... 29

3.3.2 Pufahl et al. (1983) ...................................................................................... 33

3.3.3 Vanapalli & Oh (2012) ............................................................................... 36

CHAPTER 4 – Estimating the Critical Height of Unsupported Vertical Trenches in

Sand ........................................................................................................ 39

4.1 Soil Properties .................................................................................................... 40

4.2 Estimating the Critical Height with EREPT ...................................................... 44

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4.3 Estimating the Critical Height with BSM .......................................................... 51

4.4 Comparison of Critical Heights from EREPT and BSM ................................... 58

4.5 Summary and Conclusions ................................................................................. 61

CHAPTER 5 – Estimating the Critical Height of Unsupported Trenches with

Different Wall Slopes in Sand .............................................................. 62

5.1 Estimating the Critical Height with the Finite Element Approach .................... 63

5.1.1 Hydraulic Conductivity Function ............................................................... 63

5.1.2 Analysis in SIGMA/W ................................................................................ 65

5.1.3 Analysis in SLOPE/W ................................................................................ 67

5.2 Estimating the Critical Height with the Limit Equilibrium Method .................. 70

5.3 Comparison of Limit Equilibrium and Finite Element Approaches .................. 74

5.4 Summary and Conclusions ................................................................................. 86

CHAPTER 6 – Estimating the Critical Height of Unsupported Vertical Trenches

Subjected to Surcharge Pressure ......................................................... 88

6.1 Soil Properties .................................................................................................... 88

6.2 Estimating the Critical Height without Foundation Stress ................................. 90

6.2.1 Estimating the Critical Height with EREPT ............................................... 90

6.2.2 Estimating the Critical Height with BSM ................................................... 95

6.2.3 Comparison of Critical Heights from EREPT and BSM ............................ 97

6.3 Estimating the Critical Height with Foundation Stress ...................................... 99

6.4 Summary and Conclusions ............................................................................... 107

CHAPTER 7 – General Conclusions........................................................................... 108

7.1 Recommendations for Future Research ........................................................... 111

REFERENCES .............................................................................................................. 112

CURRICULUM VITAE

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LIST OF TABLES

Table 2.1. Summary of the maximum allowable height of an unsupported vertical trench

............................................................................................................................................. 6

Table 2.2. Total cost breakdown for pipe-laying crew (Gulf Operators 2018) ................ 17

Table 4.1. Basic soil properties of Unimin 7030 sand (Mohamed & Vanapalli 2006) .... 41

Table 4.2. Summary of fitting parameters used to plot the SWCC of Unimin 7030 sand

........................................................................................................................................... 42

Table 4.3. Variation of the critical height in Unimin 7030 sand ...................................... 58

Table 6.1. Basic soil properties of IHT (Vanapalli et al. 1997, Oh & Vanapalli 2010) ... 89

Table 6.2. Variation of the critical height in IHT ............................................................. 97

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LIST OF FIGURES

Figure 2.1. Factors affecting trench stability (Ontario 2017) ............................................ 9

Figure 2.2. Excavation adjacent to existing foundations (Ontario 2017) .......................... 9

Figure 2.3. Trench parallel with backfill material (Ontario 2017) ................................... 10

Figure 2.4. Sequential trench failure (Alberta 2009) ....................................................... 11

Figure 2.5. Trench failure modes (Manitoba 2011) ......................................................... 12

Figure 2.6. Sloping standards (Ontario 2017) .................................................................. 14

Figure 2.7. Placement of spoil pile (Ontario 2017) .......................................................... 14

Figure 2.8. Shored trench with partially sloped walls (Ontario 2017) ............................. 15

Figure 2.9. Typical shored trench (Ontario 2017) ........................................................... 16

Figure 3.1. Surface tension phenomenon at air-water interface; (a) intermolecular forces

acting on contractile skin and (b) surface tension forces associated with curved 2D

surface (Fredlund et al. 2012) ........................................................................................... 19

Figure 3.2. Typical SWCC showing different zones of desaturation (Nishimura et al.

2008) ................................................................................................................................. 20

Figure 3.3. Pore-water profiles in the vadose zone (Fredlund & Rahardjo 1993a) ......... 21

Figure 3.4. Typical SWCC for silt soil (Fredlund et al. 2012) ........................................ 22

Figure 3.5. Extended Mohr-Coulomb failure envelope for unsaturated soil (Fredlund &

Rahardjo 1993b)................................................................................................................ 25

Figure 3.6. Variation of shear strength with respect to suction for four sands (modified

after Donald 1957) ............................................................................................................ 26

Figure 3.7. Relationship between and Ip (Garven & Vanapalli 2006) .......................... 28

Figure 3.8. Soil structure system for the Rankine solution for = 90° (modified after

Bowles 2001) .................................................................................................................... 30

Figure 3.9. General conditions of Mohr’s circle to derive the Rankine earth pressure

equations (modified after Bowles 2001) ........................................................................... 31

Figure 3.10. AEP and critical height in saturated soil (modified after Pufahl et al. 1983)

........................................................................................................................................... 32

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Figure 3.11. Lateral earth pressure states for saturated and unsaturated conditions (Pufahl

et al. 1983) ........................................................................................................................ 33

Figure 3.12. Components of AEP distribution and critical height in unsaturated soil .... 35

Figure 3.13. Unsupported vertical trench (unsaturated, intact, modified after Pufahl et al.

1983) ................................................................................................................................. 36

Figure 3.14. Field tests on the stability of an unsupported excavation in an unsaturated

soil (modified after Whenham et al. 2007) ....................................................................... 37

Figure 4.1. Measured matric suction profile and assumed hydrostatic matric suction

profile with the water table at a depth of 600-mm from the soil surface (modified after

Mohamed & Vanapalli 2006) ........................................................................................... 40

Figure 4.2. Grain size distribution curve of Unimin 7030 sand ....................................... 41

Figure 4.3. SWCC of Unimin 7030 sand (modified after Mohamed & Vanapalli 2006) 42

Figure 4.4. Direct shear test results on Unimin 7030 sand (modified after Mohamed &

Vanapalli 2006) ................................................................................................................. 43

Figure 4.5. Example of regression curve for the net AEP distribution with D = 0.6-m .. 45

Figure 4.6. Positive, negative, and net AEP distribution (D = 0.3-m) ............................. 47




Figure 4.10. Forces acting on the ith slice in Bishop’s simplified method of slices (Craig

2004) ................................................................................................................................. 51

Figure 4.11. Establishing PWP profile with In-situ analysis in SIGMA/W .................... 53

Figure 4.12. Analysis tree for Bishop’s simplified method with In-situ as the parent

analysis .............................................................................................................................. 53

Figure 4.13. Calculation of PWP based on the location of GWT in SIGMA/W ............. 54

Figure 4.14. Schematic of the ‘Entry and Exit’ slip surface method in SLOPE/W ......... 55

Figure 4.15. Slope stability analysis in SLOPE/W using Bishop’s simplified method

(D = 0.3-m) ....................................................................................................................... 56


(D = 0.6-m) ....................................................................................................................... 56

ix


(D = 1.0-m) ....................................................................................................................... 57


(D = 1.5-m) ....................................................................................................................... 57

Figure 4.19. Variation of the critical height with respect to the depth of the GWT using

extended Rankine earth pressure theory and Bishop’s simplified method (Unimin 7030

sand) .................................................................................................................................. 60

Figure 5.1. Different types of trenches considered in this chapter .................................. 63

Figure 5.2. Hydraulic conductivity function of Unimin 7030 sand ................................. 65

Figure 5.3. Boundary conditions required for coupled analyses in SIGMA/W ............... 67

Figure 5.4. Example of slope stability analysis tree used in the coupled analyses (10-s

time steps between excavations) ....................................................................................... 68

Figure 5.5. Example of slope stability analysis using SIGMA-Stress method in

SLOPE/W (1.5V:1H) ........................................................................................................ 69

Figure 5.6. Example of half-sine function used in M-P analyses (λ = 0.25) .................... 71

Figure 5.7. Forces acting on a slice within an arbitrary slip surface (GEO-SLOPE 2015)

........................................................................................................................................... 72

Figure 5.8. Variation of deformation, PWP, and FOS with time for (a) 10-s, (b) 250-s, (c)

500-s, and (d) 750-s after 1.3-m excavation stage with initial D = 0.7-m ........................ 76

Figure 5.9. FOS vs. Time for 1.3-m excavation stage (1.5V:1H) with initial D = 0.7-m 76

Figure 5.10. Variation of the critical height for different sloping angles in Unimin 7030

sand (10-s time steps, coupled analysis) ........................................................................... 78


sand (1,000-s time steps, coupled analysis) ...................................................................... 79


sand (M-P method) ........................................................................................................... 79

Figure 5.13. Variation of the critical height for 90° slope in Unimin 7030 sand ............ 80

Figure 5.14. Variation of the critical height for 3V:1H slope in Unimin 7030 sand ....... 80

Figure 5.15. Variation of the critical height for 2V:1H slope in Unimin 7030 sand ....... 81

Figure 5.16. Variation of the critical height for 1.5V:1H slope in Unimin 7030 sand .... 81

x

Figure 5.17. Distribution of total normal stress during 0.3-m excavation stage for

1.5V:1H slope with initial D = 1.2-m (10-s time steps) ................................................... 84

Figure 5.18. Distribution of shear stress during 0.3-m excavation stage for 1.5V:1H slope

with initial D = 1.2-m (10-s time steps) ............................................................................ 84

Figure 5.19. Vertical trench with D = 0.9-m (M-P method) ............................................ 85

Figure 5.20. Partially sloped (1:1) vertical trench with D = 0.9-m (M-P method) .......... 85

Figure 5.21. Variation of the critical height for 90*** slope in Unimin 7030 sand ........ 86

Figure 6.1. Grain size distribution curve of IHT (Oh &Vanapalli 2010) ........................ 89

Figure 6.2. SWCC of IHT (Oh & Vanapalli 2018) .......................................................... 90

Figure 6.3. Positive, negative, and net AEP distribution (D = 0-m) ................................ 91





(D = 0-m) .......................................................................................................................... 95


(D = 1-m) .......................................................................................................................... 95


(D = 3-m) .......................................................................................................................... 96


(D = 5-m) .......................................................................................................................... 96

Figure 6.11. Variation of the critical height with respect to the depth of the GWT using

extended Rankine earth pressure theory and Bishop’s simplified method (IHT) ............. 98

Figure 6.12. SLOPE/W model used to consider foundation stress ................................ 100

Figure 6.13. Stability analysis considering 50-kPa surcharge at a distance of 3-m

(D = 5-m) ........................................................................................................................ 101

Figure 6.14. Critical height vs. distance of foundation stress from excavation (D = 0-m)

......................................................................................................................................... 101


......................................................................................................................................... 102

xi


......................................................................................................................................... 102


......................................................................................................................................... 103


......................................................................................................................................... 103

Figure 6.19. Critical height vs. depth of the GWT with distance of foundation stress from

excavation = 0-m ............................................................................................................ 105


excavation = 1-m ............................................................................................................ 105


excavation = 2-m ............................................................................................................ 106


excavation = 3-m ............................................................................................................ 106

xii

LIST OF SYMBOLS & ABBREVIATIONS

Abbreviations

AEP = active earth pressure

AEV = air-entry value

BSM = Bishop’s simplified method

CW = constant water content triaxial test

EREPT = extended Rankine earth pressure theory

FOS = factor of safety

GWT = groundwater table

IHT = Indian Head till

LEM = limit equilibrium method

M-P = Morgenstern-Price

MESA = modified effective stress approach

MTSA = modified total stress approach

PAP = pore-air pressure

PWP = pore-water pressure

SWCC = soil-water characteristic curve

Symbols

= inclination of slice base (°)

wall inclination (°)

= length of slice base (m)

backfill inclination (°)

= unit weight of soil (kN/m3)

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= normalized water content

= volumetric water content

failure angle

= the first derivative of the SWCC with respect to suction

= fitting parameter for shear strength

= scaling factor for specified force function used in the Morgenstern-Price method

= Poisson’s ratio

= total stress (kPa)

= effective stress (kPa)

au = net normal stress (kPa)

= shear strength (kPa)

= internal friction angle (˚)

b = angle describing rate of change between suction and shear strength (˚)

= soil specific parameter for effective stress in unsaturated soil

= soil suction (kPa)

, , a m n = fitting parameters for Fredlund & Xing’s equation (1994)

C = total cohesion (kPa)

cwC = total cohesion from constant water content triaxial test (kPa)

c = effective cohesion (kPa)

D = depth of the groundwater table from the soil surface (m)

E = elastic modulus (kPa)

interslice normal force (kN/m)

xiv

e = Napier’s constant (2.71828…)

void ratio

( )f x = specified force function in the Morgenstern-Price method

sG specific gravity of soil particles

H = height (m)

pI = plasticity index

i = the interval between the range of j to N

j = the least negative pore-water pressure to be described by the final hydraulic conductivity function (kPa)

aK = Rankine’s active earth pressure coefficient

0K = Rankine’s earth pressure coefficient for at-rest conditions

k = hydraulic conductivity (m/s)

crm = ratio of the depth of the groundwater table to the critical height

N = slice base normal force (kN/m)

the maximum negative pore-water pressure to be described by the final hydraulic conductivity function (kPa)

ap = net active earth pressure (kPa)

, , R f x = geometric parameters as described in Figure 5.7

sR = radius of curvature of meniscus (m)

S = degree of saturation

rS = shear resistance (kN/m)

mS = mobilized shear force (kN/m)

sT = tensile force per unit length (kN/m)

au = pore-air pressure (kPa)

xv

wu = pore-water pressure (kPa)

u = pressure difference (kPa)

a wu u = matric suction (kPa)

( )a w bu u = matric suction at the air-entry value (kPa)

( )a w ru u = matric suction at the residual suction value (kPa)

W = weight of a slice within a sliding mass (kN/m)

X = interslice shear force (kN/m)

y = a dummy variable of integration representing the logarithm of negative pore-water pressure

tZ = depth of tension zone (m)

z = depth below soil surface (m)

Subscripts

FOS

f = factor of safety for force equilibrium

m = factor of safety for moment equilibrium

sat = unit weight of saturated soil

unsat = unit weight of unsaturated soil

w = unit weight of water

s = saturated water content

r = residual water content

xvi

1 = major principal stress (kPa)

3 = minor principal stress (kPa)

h = lateral pressure (kPa)

v, z = overburden pressure (kPa)

sat = shear strength of saturated soil

unsat = shear strength of unsaturated soil

H

cr = critical height of a trench

max = maximum negative pressure head

sat = height of saturated soil layer

unsat = height of unsaturated soil layer

k

sat = the measured hydraulic conductivity for saturated conditions (m/s)

unsat = the calculated hydraulic conductivity for unsaturated conditions (m/s)

1

CHAPTER 1

INTRODUCTION

1.1 Problem Statement

Most engineering projects involving foundations, landfills, pipelines, storm drains, etc., are

initiated with an excavation for infrastructure to be installed. Workers are often required to

enter the excavation or trench during the construction process, which may present serious

risks. Trenching is inherently dangerous due to the risk of cave-in, which may result in

severe injury, death, and/or consequential damage to adjacent property. Thousands of

work-related deaths and injuries in the construction industry have been attributed to trench

collapses. An average of 50 fatalities were reported each year from 1992 to 2007 in the

U.S. alone (NIOSH 2013). In Canada, each province enforces strict regulations with

respect to safe excavation practices in an attempt to prevent fatalities and serious injuries

resulting from trench collapses. The regulations specify the maximum allowable height of

an unsupported vertical trench (i.e. safe height), maximum sloping and benching angles,

and the minimum distance from the trench for stockpiling excavated or backfill materials.

Regardless of in-situ field conditions, Canadian provinces enforce safe heights in the range

of 1.2-m (New Brunswick 1991) to 1.5-m (Alberta 2009).

Trench boxes are often a practical solution in protecting workers since they allow them to

safely access the work space. However, typical trench boxes are eight metres in length and

weigh multiple tons (Trench Shoring Company 2016). Hence, the process of lifting a box,

setting it in place, and repositioning it when necessary can be very time consuming. They

2

also create an obstacle for the workers and the equipment that are used for setting

infrastructure. Lost production time results in lost profit, making it desirable to excavate

and work in unsupported vertical trenches whenever possible. However, unsupported

trenches risk collapsing and therefore must be designed and excavated with extreme

precaution, especially when workers are required to enter the excavation.

The critical height (i.e. maximum depth of a trench that can be excavated without failure,

Hcr) is the most important design consideration for ensuring the stability of unsupported

trenches. Many construction projects involve trenching and setting infrastructure in the

vadose zone; thus, the critical height of unsupported trenches should be determined by

extending the mechanics of unsaturated soils. Trench stability is mainly governed by the

matric suction distribution between the soil surface and the groundwater table (Pufahl et al.

1983, Whenham et al. 2007, De Vita et al. 2008, Vanapalli & Oh 2012). In other words,

adhering to a universal safe height suggested by provincial regulations may not be a

reasonable approach in geotechnical engineering practice, because in-situ field conditions

are not considered.

1.2 Objectives of the Thesis

The main objective of this study is to investigate the variation of the critical height of

unsupported trenches in unsaturated soil considering various field conditions and matric

suction distributions. More details are as follows:

- Review current trenching regulations and procedures in Canada.

- Review methodologies for estimating the critical height of unsupported vertical

trenches in unsaturated soil.

3

- Estimate the critical height of unsupported trenches in unsaturated cohesionless

(Unimin 7030 sand) and cohesive (Indian Head till) soils using analytical and

numerical approaches.

1.3 Scope of the Thesis

Multiple excavation scenarios were simulated considering five major factors; i) level of the

groundwater table (hereafter referred to as GWT) ii) slope of the trench wall, iii) excavation

rate, iv) magnitude of surcharge pressure, and v) distance between a trench and a surcharge

pressure. The critical heights were estimated using three distinct approaches; i) analytical

approach with extended Rankine earth pressure theory, ii) limit equilibrium methods

(Bishop’s simplified method and Morgenstern-Price method), and iii) slope stability

analyses based on finite element analyses. Geotechnical modelling software, SLOPE/W

and SIGMA/W (GEO-SLOPE International Ltd.), were used for the limit equilibrium

methods and finite element analyses. The critical heights obtained from the three different

approaches are discussed. The results were also compared to the guidelines set by Canadian

provinces. Lastly, recommendations were provided for the direction of future studies in

estimating the critical height of unsupported trenches.

1.4 Organization of the Thesis

This thesis consists of seven chapters including ‘Introduction’ (Chapter 1) and ‘General

Conclusions’ (Chapter 7).

4

Chapter 2 discusses trenching regulations and general trenching guidelines in Canada. The

modes and mechanisms of trench failure are described, and field procedures for mitigating

worker risks are discussed. Trenching economics are briefly noted.

Chapter 3 is an extensive literature review pertaining to the mechanical properties of

unsaturated soil and analytical approaches for estimating the critical height of unsupported

vertical trenches in unsaturated soil.

Chapter 4 details the procedures to estimate the critical height of unsupported vertical

trenches in an unsaturated sand using two independent methods; i) extended Rankine earth

pressure theory and ii) Bishop’s simplified method, which is a form of the Limit

Equilibrium Method (hereafter referred to as LEM). The critical heights obtained from both

approaches are compared. In addition, details of the geotechnical modelling software,

SLOPE/W and SIGMA/W, are provided.

Chapter 5 investigates the influence of excavation rate on the critical height of various

sloped trenches in an unsaturated sand through coupled stress-PWP analyses, which is a

finite element approach. The estimated critical heights for different time steps (i.e. fast and

slow excavation rates) are compared with those from the Morgenstern-Price method, which

is also a form of the LEM. Additional details of SIGMA/W are provided.

Chapter 6 examines the critical height of unsupported vertical trenches in an unsaturated

cohesive soil using the LEM (Morgenstern-Price method). The trenches being modelled

are subjected to surcharges pressures at various distances from the edge of excavation.

5

CHAPTER 2

TRENCHING HAZARDS AND REGULATIONS IN CANADA

‘Trench’ is defined as an excavation in which the measurement of depth exceeds the width.

Working in trenches is inherently dangerous because workers are subjected to the risk of

cave-ins. According to NIOSH (2013), trenches as shallow as 1.2-m deep have reportedly

collapsed and resulted in deaths. To avoid these types of tragedies, each Canadian province

enforces strict regulations with respect to safe excavation practices. Table 2.1 summarizes

the maximum allowable height of an unsupported vertical trench as regulated by each

Canadian province before safety measures such as benching, sloping, bracing, or trench

boxes must be implemented to access the workspace (i.e. safe height). The problem with

enforcing a universal safe height throughout each province is that the standards are solely

based on empirical data without any theoretical background, rather than considering the

soil type and field conditions. More importantly, the factor of safety used in determining

the safe height in each province is unknown. This may lead to over, or under conservative

protective measures being used during construction. De Vita et al. (2008) reported that

stable vertical cuts can be approximately 15-m deep in a Pozzolan deposit, while Richard

et al. (2017) concluded that the critical height in an unsaturated sand can be even less than

1-m. These studies indicate that the safe height of an unsupported trench should be

estimated based on the job site conditions.

6

Table 2.1. Summary of the maximum allowable height of an unsupported vertical trench

Provinces in Canada Safe height

BC, NB, NS, ON, QC, SK (NL, PE) 1.2-m (4-ft)

AB, MB 1.5-m

2.1 Trench Collapse Statistics

According to NIOSH (2013), an average of 50 fatalities were reported annually in the U.S.

from 1992-2007 due to trench collapses. Eivemark & Hall (2000) stated that approximately

1% of work-related deaths in the U.S. were caused by trenching accidents. Other studies

showed that approximately 1,000 injuries, and between 60 and 100 deaths per year were

associated with trench failures in the U.S. (Thompson & Tanenbaum 1977, Suruda et al.

1988, White 2008). Most deaths occurred in sewer line construction when the trenches

were not shored. 80% of reported fatalities occurred in trenches less than 4.5-m deep, and

40% occurred in trenches less than 3-m deep (Eivemark & Hall 2000). Shoring or trench

boxes normally provide sufficient worker protection. However, workers should exit the

trench before shoring or a trench box is moved (Hinze 2005).

The number of deaths and injuries related to trenching have significantly declined as

regulations have been improved. In New Brunswick, the last death that was caused by

trenching was reported on May 28, 2001 (WorkSafeNB 2017). The employer was charged

with non-compliance of regulations 181(1) and 182(1)(a) of the Occupational Health &

Safety Act (1991), which are listed below:

7

181(1): An employer shall ensure that the walls of an excavation or trench are

supported by shoring, bracing, or caging, except when the excavation or trench

(a) is less than 1.2-m deep,

(b) subject to subsection (2), is cut in solid rock,

(c) is sloped or benched to within 1.2-m of the bottom of the excavation or trench

with the slope or bench not exceeding 1-m of vertical rise to each 1-m of

horizontal run, or

(d) is one that an employee is not required to enter.

181(2): Where the walls or crests of an excavation or trench are cut in solid rock

and are not stable, an employer shall ensure that the walls and crests are adequately

supported by rock bolts, wire mesh, shoring, or a method that provides equivalent support.

182(1): An employer shall ensure that an employee does not, and no employee shall,

enter an excavation or trench 1.2-m or more in depth unless

(a) the walls of the excavation or trench are supported by shoring, bracing, or

caging, the excavation or trench is cut in solid rock, or the excavation or trench

is sloped or benched to within 1.2-m of the bottom of the excavation or trench

with the slope not exceeding 1-m of vertical rise to each 1-m of horizontal run.

Only infraction codes were provided in the investigation report without site-specific details.

The employer was fined $4,800 for their negligence.

8

2.2 Trench Failure Mechanisms

Changes in pore-water pressure (hereafter referred to as PWP), surface loading, and

vibration are the most predominant causes of trench instability (Ontario 2017). Figure 2.1

illustrates the causes of trench failure.

The base and walls of a trench experience elastic rebound immediately after excavating

due to the relief of confining pressures. This causes an increase in the soil’s void space and

a decrease in PWP, thus increasing the soil’s shear strength. Hence, a trench may appear

stable immediately after excavating. However, the trench destabilizes over time as PWP

rises back to equilibrium within the soil nearby the trench base and walls. Rainfall

infiltration, development of cracks, and a rising GWT can further increase PWP and

accelerate a trench failure. Thus, the risk of trench collapse increases as exposure time to

the atmosphere increases. Trenches should be backfilled as soon as the job is complete.

Many forms of static and dynamic loading can cause a trench to collapse. Surcharge loads

from excavated material and construction equipment, and vibration induced by vehicular

traffic, compaction, pile driving, blasting, other adjacent excavations, etc. may trigger a

collapse. Trenches excavated near existing foundations may collapse if the bearing

pressure enters the active zone of the trench (Figure 2.2).

9

Figure 2.1. Factors affecting trench stability (Ontario 2017)

Figure 2.2. Excavation adjacent to existing foundations (Ontario 2017)

The presence of buried utilities adjacent to an open excavation can result in an unexpected

failure because the backfill materials encasing the utility are typically weaker than

undisturbed native soil (Figure 2.3). This type of failure mechanism not only endangers

workers, but may also result in excessive strain causing structural damage to the utilities

10

(Symons et al. 1982). Shoring is typically required when excavating near buried utilities or

foundation structures to ensure ground movement is minimized.

Figure 2.3. Trench parallel with backfill material (Ontario 2017)

Heterogeneity should also be monitored when excavating, especially over long distances.

Some soils such as gravel deposits may have a high variance with respect to local

mechanical properties (Krahn & Fredlund 1983, Dai et al. 2016). Thus, the critical height

may vary significantly between locations in the same soil deposit. In such cases, it may be

necessary to perform a probabilistic analysis to determine minimum values for the soil

properties of interest to quantify a critical height.

2.3 Trench Failure Modes

The Occupational Health & Safety Division of Alberta (2009) described how rescue

attempts may be more difficult and dangerous when the wall failure occurs sequentially as

shown in Figure 2.4. In this case, failure is initiated at the base of the trench wall. Cracking

near the ground surface and local failures in Zone 1 should raise alarms. This localized

failure or movement leads to failure in Zone 2. Finally, the failure in Zone 3 occurs due to

the self-weight of the soil. This failure sequence is a plausible explanation for why rescuers

11

are sometimes trapped along with the first victim(s). Someone attempting to intervene and

help uncover a victim when failure in Zone 3 has not yet occurred can put themselves at

risk.

Figure 2.4. Sequential trench failure (Alberta 2009)

The Workplace Safety & Health Division of Manitoba (2011) categorizes the modes of

trench collapse into four types (Figure 2.5).

(a) Spoil pile slide - occurs when the excavated material is not placed far enough

away from the edge of the excavation. A minimum distance of 0.6-m is

recommended for every one-metre of excavation depth.

(b) Side wall shear - common to fissured or desiccated clay-type or alluvial soils that

are exposed to drying.

(c) Slough-in (cave-in) - common to previously excavated material, fill, and granular

soils where the water table is above the base of the excavation, or where soils are

organic or peat.

(d) Rotation - common in clay-type soils when excavation walls are too steep, or

when moisture content increases rapidly.

12

Figure 2.5. Trench failure modes (Manitoba 2011)

2.4 Work Protection Methods

At least one of benching, sloping, shoring, or worker-protective trench boxes must be

implemented in the trenching process when it is necessary to excavate deeper than the

regulated safe height. The preferred safety measure depends on the application and the soil

properties. Protection methods have been devised based on four soil types (Figure 2.6;

Ontario 2017):

(a) Type 1 - includes hardpan, consolidated clay, and some glacial tills. It is hard to

drive a pick into Type 1 soil. The sides remain vertical with no water released

from the trench wall during excavation. Rain may cause soil to break down along

the edges of the excavation.

(a) (b)

(c) (d)

13

(b) Type 2 - Includes silty clay and less dense tills. It is relatively easy to drive a pick

into Type 2 soil and can be easily excavated with a backhoe. The sides will

remain vertical for short periods of time before tension cracks appear as the soil

starts to dry.

(c) Type 3 - Includes sand, granular materials, silty or wet clays, and all backfill or

previously disturbed material. Type 3 soil can be excavated without difficulty

using a backhoe and will not stand vertically when dry. Soil will stand vertically

for short periods of time when wet, however it dries quickly and is prone to

deterioration by vibration from heavy equipment.

(d) Type 4 - Includes quicksand, Leta clays, silty clays, muskeg, or other organic

deposits with high moisture content. Type 4 soil is flowable material that can be

excavated with no difficulty and must be supported or contained to excavate to

any significant depth.

Regardless of the material, the spoil pile should be placed further away from the edge of

excavation as depth increases (Figure 2.7; Ontario 2017). It is necessary to slope the trench

walls 1:1 once the excavation extends deeper than the safe height in Type 1 and 2 soils.

Type 3 and 4 soils warrant 1:1 and 1V:3H slopes from the base of the excavation,

respectively (Figure 2.6). The main restriction with sloping is the space allotted to achieve

an adequate slope, and therefore other reinforcement may be required in certain practical

scenarios. Figure 2.8 illustrates the case of using shoring in conjunction with sloping.

14

Figure 2.6. Sloping standards (Ontario 2017)

Figure 2.7. Placement of spoil pile (Ontario 2017)

Good soil

Faily good soil Type 3 soil

Maximum1.2-m

Bad soil Type 4 soil

Type 1 & 2 soil

Minimumbank slope

Minimumbank slope

Minimumbank slope

15

Figure 2.8. Shored trench with partially sloped walls (Ontario 2017)

Shoring involves supporting trench walls with a braced structure as shown in Figure 2.9.

Shoring systems are used to transfer and resist the loads acting on opposing sides of the

trench. Hydraulic shoring is more frequently used in practice than timber shoring (Hinze

2005). Hydraulic shores are prefabricated sections in which the width can be adjusted with

a pneumatic device, making them much more practical than timber shores that must be cut

to the exact width of the excavation. Trench boxes are different from shores as they do not

provide structural support to trenches unless the void space between the box and the walls

are backfilled; their sole purpose is to prevent soil from collapsing onto the workers. For

this reason, trench boxes should always be accessed by a ladder which rests inside the box.

The space between the box and the wall should be minimized to provide easier access to

the ladder from the edge of the excavation, and to limit soil movement in the event of a

collapse. It is often necessary to use shoring rather than trench boxes while working in the

proximity of infrastructure such as buried utilities, roads with nearby traffic, or buildings.

16

Figure 2.9. Typical shored trench (Ontario 2017)

2.5 Trench Economics

Unsupported trenches require the least amount of excavation work and are therefore the

most desirable form of trenches from an economical perspective. Benching and sloping

require more time for excavating since more soil is removed per linear distance, yet the

installation process may be unimpeded if the infrastructure cannot be installed as fast as

the trench is being excavated. However, production is inevitably slowed when using

shoring or trench boxes. An anonymous construction company was capable of setting 6 to

8 (0.6-m diameter, 6-m long) pipes per day when working in and around a trench box,

compared to 12 to 15 per day when working without one. Inconsistencies in day-to-day

operations were mostly attributed to human resources, and trench dewatering associated

17

with groundwater seepage or rainfall. The opportunity cost of employing an entire pipe-

laying crew with heavy equipment for a 12-hour day is approximately $10,000. A total cost

breakdown is shown below in Table 2.2. Other costs that are not listed in Table 2.2 include

the cost of quality control inspectors, surveyors, and rental equipment. Over an entire

construction season on a large-scale project, hundreds of thousands of dollars could be

saved if working without a trench box or shoring could be justified.

Table 2.2. Total cost breakdown for pipe-laying crew (Gulf Operators 2018)

Equipment Cost per hour ($CAD/hr)

Quantity Cost per 12-hour day ($CAD/day)

Cat 345 Excavator 215.00 1 2,580.00

Cat 336 Excavator 190.00 1 2,280.00

Cat D-4 Dozer 95.00 1 1,140.00

Compactor 85.00 1 1,020.00

Labour 37.00 4 1,776.00

Lead Hand 41.00 1 492.00

Supervisor 68.00 1 816.00

Total cost per day 10,104.00

18

CHAPTER 3

THEORETICAL BACKGROUND

3.1 Properties of Unsaturated Soil

Unsaturated soil consists of soil particles, water, and air (i.e. solid, liquid, and gas phases).

The three phases interact to create the contractile skin (or air-water interface), which is

referred to as the fourth phase. As shown in Figure 3.1(a), a molecule of water within a

body experiences equal forces in all directions, while a molecule along the air-water

interface experiences an unbalanced force towards the interior of the water body. Thus, a

tensile force is generated tangentially to the air-water interface to satisfy static equilibrium.

Surface tension causes the contractile skin to behave like an elastic membrane; somewhat

like an inflated balloon that has a greater air pressure inside than in the atmosphere. The

membrane forms a concave curvature towards the larger pressure (Figure 3.1(b)). The

radius of curvature is directly related to the pressure difference as shown in Eq. (3.1). The

tensile force is temperature dependant. In an unsaturated soil, the air pressure (ua = 0-gauge

pressure) becomes greater than the water pressure (uw = negative gauge pressure) which

creates a contractile skin. The pressure difference between ua and uw is referred to as matric

suction (Eq. (3.2)).

2 s

s

Tu

R (3.1)

19

2 sa w

s

Tu u

R (3.2)

where Δu = pressure difference, Ts = tensile force per unit length, Rs = radius of curvature,

and (ua – uw) = matric suction

Figure 3.1. Surface tension phenomenon at air-water interface; (a) intermolecular forces acting on contractile skin and (b) surface tension forces associated with curved 2D surface (Fredlund et al. 2012)

Eq. (3.1) shows that matric suction approaches infinity as the radius of curvature

approaches zero, indicating that smaller radii can exert greater matric suction than larger

radii. In terms of the radii within the void space of an unsaturated soil, fine-grained soils

with small pore-sizes have a higher affinity for matric suction than coarse-grained soils

(a)

(b)

20

with relatively large pores. In other words, the ability for a soil to retain water is strongly

related to the pore-size (or particle-size) distribution.

The soil-water characteristic curve (hereafter referred to as SWCC) is a soil specific

relationship between matric suction and the degree of saturation (or volumetric water

content). A typical SWCC can be divided into three main zones; boundary effect zone,

transition zone, and residual zone (Figure 3.2).

Figure 3.2. Typical SWCC showing different zones of desaturation (Nishimura et al. 2008)

The boundary effect zone ranges from zero matric suction (i.e. saturation) up to the air-

entry value (hereafter referred to as AEV). The AEV is the suction required to force air

into the largest voids in a soil and is the first point of importance on a SWCC for a drying

Suction (kPa)

1 10 100 1000 10000 100000 1000000

Deg

ree

of

satu

rati

on

, S (

%)

0

20

40

60

80

100

Boundaryeffect zone

Transitionzone

Residual zoneof

unsaturation

Air-entry value(ua - uw)b

Residualsuction value

Air

Soil

Water film

21

cycle. Soil begins desaturating once matric suction is increased past the AEV. Most

desaturation occurs in the transition zone. The degree of saturation (or water content) is

reduced significantly with relatively small increases in matric suction. In the residual zone,

large increases in suction result in small changes in the degree of saturation.

Figure 3.3 demonstrates possible PWP profiles with respect to depth above and below the

GWT. In the saturated zone, the pore-water is in compression (i.e. zero matric suction) and

the magnitude of pressure increases linearly with depth. In the unsaturated zone (or vadose

zone), the pore-water is in tension and the soil is subjected to a matric suction distribution

that can vary linearly or nonlinearly above the GWT (Fredlund & Rahardjo 1993a). Ideal

hydrostatic conditions (i.e. linear matric suction distribution; ① in Figure 3.3) often do not

represent true field conditions due to environmental factors such as evaporation (② in

Figure 3.3) and rainfall infiltration (③ in Figure 3.3).

Figure 3.3. Pore-water profiles in the vadose zone (Fredlund & Rahardjo 1993a)

22

Soils in nature experience cycles of wetting and drying (i.e. rainfall infiltration and

evaporation), which alter the SWCC of the soil accordingly. Increasing or decreasing a

soil’s moisture content by the same volume of water does not cause the same change in

suction. This phenomenon is known as hysteresis. The first drying (desorption curve) and

wetting (adsorption curve) cycle form the two boundaries for the possible SWCCs that a

soil can experience (Figure 3.4). Larger pores drain first due to better interconnectivity,

while smaller pores are more reluctant to release water when drying since there is more

surface contact between the available water and the soil grains. Conversely, larger pores

are filled last during wetting because smaller pores absorb water more easily due to a higher

affinity for matric suction. Thus, the drying SWCC always corresponds to higher suction

than the wetting SWCC for a given moisture content (Yang et al. 2004, Goh et al. 2014).

It is therefore more conservative to analyze the mechanical behaviours of unsaturated soils

using a wetting-cycle SWCC.

Figure 3.4. Typical SWCC for silt soil (Fredlund et al. 2012)

1,000,000

23

3.2 Shear Strength of Unsaturated Soil

The effective stress equation proposed by Terzaghi (1936) laid a foundation towards

establishing geotechnical engineering as one of the independent fields of civil engineering

(Vanapalli 2009). Effective stress is the difference between total stress and PWP (Eq. (3.3)).

In conjunction with Mohr-Coulomb theory, effective stress can be used to interpret the

shear strength of saturated soils (Eq. (3.4)).

wu (3.3)

tansat c (3.4)

where ’ = effective stress, = total stress, uw = pore-water pressure, sat = shear strength

of saturated soil, c’ = effective cohesion, and ’ = effective internal friction angle

Most soils, especially in arid or semi-arid regions, exist in unsaturated conditions. In this

case, the influence of matric suction should be considered to reliably estimate the shear

strength of unsaturated soil. Bishop (1959) was one of the first researchers to recognize

that suction contributes to the strength of unsaturated soil and proposed Eq. (3.5) by

extending Terzaghi’s equation for effective stress (i.e. Eq. (3.3)).

a a wu u u (3.5)

where ua = pore-air pressure, ( – ua) = net normal stress, and = a soil parameter related

to degree of saturation (or matric suction) ranging from 0 to 1

24

The effective stress parameter, χ, is equal to zero for dry soil. When the soil is saturated, χ

is equal to one and Eq. (3.5) becomes equal to Eq. (3.3). Jennings & Burland (1962) and

Bishop & Blight (1963) pointed out that there are limitations to Eq. (3.5) since different χ

values are required to explain shear strength and volume change behaviour of unsaturated

soils. In other words, Eq. (3.5) cannot universally describe the mechanical behaviours of

unsaturated soil. The parameter, , can be evaluated using Eq. (3.6) to determine the shear

strength of unsaturated soils depending on the range of matric suction values (Khalili &

Khabbaz 1998).

1

a wa w a w b

a w b

a w a w b

u uif u u u u

u u

if u u u u

(3.6)

where (ua – uw)b = air-entry value of a soil

Fredlund & Morgenstern (1977) concluded that net normal stress and matric suction are

the two independent stress-state variables that are to be used for evaluating the mechanical

behaviour of unsaturated soil. Fredlund et al. (1978) proposed Eq. (3.7) to estimate the

shear strength of unsaturated soil based on the extended Mohr-Coulomb failure envelope

(Figure 3.5).

tan tan bunsat a a wc u u u (3.7)

where unsat = shear strength of unsaturated soilb = angle indicating the rate of increase in

shear strength with respect to a change in matric suction

25

Figure 3.5. Extended Mohr-Coulomb failure envelope for unsaturated soil (Fredlund & Rahardjo 1993b)

However, determining b is time-consuming and requires elaborate testing equipment.

Previous research also showed that b is not a constant, but varies nonlinearly with respect

to matric suction (Escario & Saez 1987, Gan et al. 1988). For matric suction values less

than the AEV, b = ’, and b becomes less than ’ once matric suction exceeds the AEV

(i.e. b < ’). It is interesting to note that shear strength approaches that of saturated

condition for suction values greater than residual suction in coarse-grained soils (Figure

3.6). This is attributed to a decrease in the net contribution of matric suction towards shear

strength as the residual condition is approached. The contractile skin becomes

discontinuous in the residual stage and becomes incapable of maintaining peak strength.

26

Figure 3.6. Variation of shear strength with respect to suction for four sands (modified after Donald 1957)

Vanapalli et al. (1996) proposed equations to estimate the variation of b with respect to

suction in terms of volumetric water content/degree of saturation as shown in Eq. (3.8) and

(3.9), respectively.

tan tanb r

s r

(3.8)

tan tan100

b r

r

S S

S

(3.9)

where S = degree of saturation, = volumetric water content, subscript s = saturated

condition, and subscript r = residual condition

Matric suction, (ua - uw) (kPa)

0 5 10 15 20 25 30 35

Sh

ear

stre

ng

th (

kPa)

5

10

15

20

25

Fine FrankstonGraded FrankstonMedium FrankstonBrown Frankston

Donald 1957

27

The SWCC for the entire range of suction is required to accurately identify the volumetric

water content or degree of saturation for a soil’s residual condition. Vanapalli et al. (1996)

proposed another equation that requires the degree of saturation (or volumetric water

content) along with a fitting parameter.

tan tan tanb S (3.10)

where = normalized volumetric water content (= /s) and = fitting parameter for the

shear strength of an unsaturated soil

Vanapalli & Fredlund (2000) provided a relationship between the fitting parameter, κ, and

the plasticity index, Ip, as shown in Eq. (3.11) using five data sets of shear strength of

unsaturated soils. Garven & Vanapalli (2006) further improved Eq. (3.11) by using ten data

sets of shear strength for compacted soils (Eq. (3.12) and Figure 3.7). Eq. (3.11) and (3.12)

show that κ = 1 for estimating the variation of shear strength of unsaturated non-plastic

soils with respect to matric suction.

20.0008( ) 0.0801( ) 1p pI I (3.11)

20.0016( ) 0.0975( ) 1p pI I (3.12)

where Ip = plasticity index

28

Figure 3.7. Relationship between and Ip (Garven & Vanapalli 2006)

Total cohesion is the sum of effective and apparent cohesion as shown in Eq. (3.13).

tan ba wC c u u (3.13)

where C = total cohesion c’ = effective cohesion, and (ua uw) tanϕb = apparent cohesion

Substituting Eq. (3.10) into Eq. (3.13), and then into Eq. (3.7) yields Eq. (3.14).

tan tan

tan tan

tan tan

tan tan

tan

bunsat a a w

a a w

a w a

a w a

a

c u u u

c u u u S

c u u S u

c u u u

C u

(3.14)

Plasticity index, Ip

0 10 20 30 40 50

Fit

tin

g p

aram

eter

,

0

1

2

3

4

5

6

7

Best fit parameter68% confidence interval

Regression

= -0.0016 (Ip2)+0.0975 (Ip)+1

A. Red Silty ClayB. Madrid Gray ClayC. Adams ClayD. Indian Head Till (dry of opt.)E. Indian Head Till (opt.)

F. Botkin SiltG. Ste. Rosalie ClayH. LD Dhanauri ClayI. HD Dhanauri ClayK. Nanyang Clay

FA

C

G

D,EBIK

H

29

3.3 Critical Height of Unsupported Vertical Trenches

3.3.1 Rankine’s Earth Pressure Theory (1857)

Rankine (1857) investigated the stress conditions of a soil in a state of plastic equilibrium.

Bowles (2001) summarized the major assumptions made in Rankine’s theory as listed

below:

(a) Soil is isotropic and homogeneous and has internal friction but no cohesion.

(b) The rupture surface (AC in Figure 3.8) is a plane surface and the backfill surface

(BC in Figure 3.8) is planar (it may slope but is not irregularly shaped).

(c) The frictional resistance is distributed uniformly along the rupture surface and

the soil-to-soil friction coefficient, f = tan.

(d) The failure wedge (ABC in Figure 3.8) is a rigid body undergoing translation.

(e) There is no wall friction.

(f) Failure is a plane strain problem - that is, consider a unit interior slice from an

infinitely long wall.

30

Figure 3.8. Soil structure system for the Rankine solution for = 90° (modified after Bowles 2001)

Plastic equilibrium is achieved if a wall moves sufficiently far away from a soil mass, in

which the effective lateral earth pressure acting on the vertical plane is denoted as

Rankine’s active earth pressure (hereafter referred to as AEP). For the case shown in Figure

3.9, AEP can be calculated using Eq. (3.15) assuming the soil’s unit weight is constant.

2 2

2 2

cos

cos cos coscos

cos cos cos

a a a

a

p z K zK

K

(3.15)

where pa = active earth pressure, γ = unit weight of soil, z = depth, β = angle of backfill,

and Ka = active earth pressure coefficient

β

θ‐β

θ

90° - θ

90° + β

ϕH

B

A

W

R

C

Pa

31

(a) General case: only for +β as shown (b) Mohr’s circle

Figure 3.9. General conditions of Mohr’s circle to derive the Rankine earth pressure equations (modified after Bowles 2001)

When the backfill surface is horizontal, the AEP coefficient can be calculated with Eq.

(3.16).

1 sin

1 sinaK

(3.16)

Rankine did not incorporate soil cohesion in estimating AEP. This issue is resolved by

combining Eq. (3.17) with Rankine’s theory as shown in Eq. (3.18).

23 1 tan 45 2 tan 45

2 2c

(3.17)

2a a ap zK c K (3.18)

where 3 = minor principal stress, 1 = major principal stress

β

β

τ

σ

σv

pa O

G

Eϕσv OG

pa OE

90° β 90° β

32

The lateral earth pressure (positive, negative, and net) distribution in a saturated cohesive

soil is shown in Figure 3.10. From the ground surface to a depth of zero net AEP is referred

to as the tension zone. The depth of the tension zone can be calculated using Eq. (3.19).

Theoretically, by setting the sum of horizontal forces equal to zero, a vertical trench can be

excavated up to twice the depth of the tension zone without failure if effective cohesion

and the unit weight are constant. In which case, the critical height of an unsupported

vertical trench is calculated using Eq. (3.20).

Figure 3.10. AEP and critical height in saturated soil (modified after Pufahl et al. 1983)

2

t

a

cZ

K

(3.19)

4

2cr t

a

cH Z

K

(3.20)

Where Hcr = critical height, Zt = depth of tension zone

H

zt

+ =

z

K -2c'K v haA

33

By assuming a planar rupture surface and a horizontal backfill surface, the AEP in an

unsupported trench in unsaturated soil can be interpreted by extending Rankine’s earth

pressure theory.

3.3.2 Pufahl et al. (1983)

Pufahl et al. (1983) investigated lateral earth pressure (i.e. active and passive pressures) for

a vertical trench extending the mechanics of unsaturated soil. Conventional Rankine earth

pressure theory for saturated soil was modified to incorporate the influence of matric

suction using the Mohr-Coulomb failure criteria (i.e. extended Rankine earth pressure

theory). The lateral earth pressures (i.e. active and passive) for both saturated and

unsaturated conditions are illustrated in Figure 3.11.

Figure 3.11. Lateral earth pressure states for saturated and unsaturated conditions (Pufahl et al. 1983)

34

The AEP decreases as a soil desaturates due to increasing contribution from matric suction.

From the geometrics of the Mohr circle in Figure 3.11, the net AEP can be written as Eq.

(3.21) by replacing effective cohesion in Eq. (3.18) with total cohesion (Eq. (3.13)) to

incorporate the influence of matric suction.

2 tan bh a v a a a w au u K c u u K (3.21)

where (h – ua) = net lateral pressure, and (v – ua) = net overburden pressure

If the pore-air pressure (hereafter referred to as PAP) is assumed to be atmospheric pressure

(i.e. ua = 0), the AEP in an unsaturated soil can be calculated using Eq. (3.22).

2 tan ba v a a w ap K c u u K (3.22)

Substituting Eq. (3.10) into Eq. (3.22) yields Eq. (3.23).

2 tan

2 tan

2

ba v a a w a

v a a w a

v a a

p K c u u K

K c u u S K

K C K

(3.23)

Typically, matric suction varies nonlinearly with depth. In other words, the critical height

in unsaturated soils may not be simply two times the depth of the tension zone. In this case,

the critical height can be estimated by locating the depth that equates ‘Area I’ and ‘Area II’

as shown in Figure 3.12.

35

Figure 3.12. Components of AEP distribution and critical height in unsaturated soil

By assuming a planar failure surface, the critical height in unsaturated soil can also be

determined by considering a balance of forces on the sliding wedge shown in Figure 3.13.

Trial and error must be used to solve for Hcr since it appears on both sides of Eq. (3.24).

2

4tan

2

2 tan1 2

bcra w

acr

wcr sat cr cr unsat

a

mc u u

KH

m m mK

(3.24)

where mcr = D/Hcr, D = depth of the GWT from the soil surface, w = unit weight of water,

and sat, unsat = unit weight of soil for saturated and unsaturated condition, respectively

36

Figure 3.13. Unsupported vertical trench (unsaturated, intact, modified after Pufahl et al. 1983)

3.3.3 Vanapalli & Oh (2012)

Vanapalli & Oh (2012) extended the work by Pufahl et al. (1983) to analyze the stability

of an unsupported vertical trench in an unsaturated soil. The excavation dimensions were

3-m deep, 6-m wide, and 20-m long. The variation of matric suction with depth was

measured using tensiometers installed at depths of 1, 1.5, 2.5, and 3.5-m from the ground

surface (Whenham et al. 2007). The trench was first excavated in June 2004. The first

localized and generalized failures were observed in January 2005 and February 2005,

respectively, due to a decrease in matric suction associated with precipitation activity

(Figure 3.14).

crH

sat

unsat

37

Figure 3.14. Field tests on the stability of an unsupported excavation in an unsaturated soil (modified after Whenham et al. 2007)

Two different approaches were used to calculate AEP; the Modified Effective Stress

Approach (hereafter referred to as MESA; Eq. (3.22)) and the Modified Total Stress

Approach (hereafter referred to as MTSA; Eq. (3.25)). The MTSA implies that excess PAP

is assumed to drain and the excess PWP is an undrained condition. Therefore, AEP is

estimated by replacing total cohesion, C, in Eq. (3.22) with total cohesion from the

constant-water content triaxial test (Ccw; Rahardjo et al. 2004, Thu et al. 2006).

2a v a cw ap K C K (3.25)

The factor of safety (hereafter referred to as FOS) estimated using the MESA and the

MTSA were 0.31 and 1.14, respectively, at the time when first localized failures occurred.

38

This implies that the stability analysis using the MESA is more conservative, but the

analysis done with constant-water content test results provides more realistic estimates

compared to the field observations. At the time general failures occurred, matric suction

remained relatively constant at depth and the wetting front remained stable even after a

small rainfall event. This indicates that general failure of the excavation can be attributed

to an increase in the AEP associated with an increase in the soil’s unit weight.

In this study, Eq. (3.23) was used to calculate the net AEP distribution for an unsupported

vertical trench in unsaturated soil.

39

CHAPTER 4

ESTIMATING THE CRITICAL HEIGHT OF UNSUPPORTED VERTICAL TRENCHES IN SAND

In this chapter, an attempt is made to estimate the critical height of an unsupported vertical

trench in an unsaturated sand. Two independent approaches were used; i) extended

Rankine’s earth pressure theory (hereafter referred to as EREPT) and ii) Bishop’s

simplified method (hereafter referred to as BSM) with geotechnical modelling software,

SLOPE/W. It was assumed that the trenches were excavated into Unimin 7030 sand with

various levels of the GWT (i.e. different matric suction distributions).

Mohamed & Vanapalli (2006) conducted model-footing tests in a sand (Unimin 7030) for

both saturated and unsaturated conditions in a specially designed soil tank. Unsaturated

conditions were achieved by setting a water level at some depth, and the matric suction

distribution was established by measuring matric suction at various depths using

conventional tensiometers. Figure 4.1 shows the measured and the assumed matric suction

distributions with the water table at a depth of 600-mm in the soil tank, in which there is a

negligible difference. Hence, a hydrostatic matric suction distribution was assumed for

both approaches used in this chapter.

40

Figure 4.1. Measured matric suction profile and assumed hydrostatic matric suction profile with the water table at a depth of 600-mm from the soil surface (modified after Mohamed & Vanapalli 2006)

4.1 Soil Properties

Basic soil properties of Unimin 7030 sand are summarized in Table 4.1. The grain size

distribution curve of the sand is shown in Figure 4.2. The SWCC was measured using a

Tempe Cell apparatus extending the axis-translation technique (Figure 4.3; Hilf 1956). The

AEV [(ua – uw)b] and residual suction value [(ua – uw)r] were estimated to be 4 kPa and 7.8

kPa, respectively. This was done using the procedure detailed in Vanapalli et al. (1999),

(i.e. the matric suction values corresponding to the intersection of the two linear slope

segments of the SWCC; Figure 4.3). A best-fit analysis for the SWCC was conducted using

Fredlund & Xing's (1994) model (Eq. (4.1)), and the fitting parameters are summarized in

Table 4.2.

10

150

300

600

500

B

1 kPa

2 kPa

4 kPa

7 kPa

Tensiometer

GWT

distribution ofmeasured matric suction

mm

mm

mm

mm

mmGWT:groundwater table

hydrostatic matric suctiondistribution

41

Table 4.1. Basic soil properties of Unimin 7030 sand (Mohamed & Vanapalli 2006)

Properties Value

Plasticity Index, Ip NP

Saturated unit weight, sat (kN/m3) 20.4

Saturated water content, θs (%) 38.7

Void ratio, e 0.63

Specific gravity, Gs 2.65

Effective cohesion, c’ (kPa) 0

Effective internal friction angle, ’ (˚) 36.2

Saturated hydraulic conductivity, ksat (m/s) 5 × 10-5

Elastic modulus, E (kPa)** 10,000

Poisson’s ratio, ν ** 0.33

** = assumed value

Figure 4.2. Grain size distribution curve of Unimin 7030 sand

Particle Size (mm)

0.01 0.1 1 10

% F

iner

0

20

40

60

80

100

Sand

42

Figure 4.3. SWCC of Unimin 7030 sand (modified after Mohamed & Vanapalli 2006)

1

ln

m

ns

Se a

(4.1)

where e = Napier’s constant (2.71828…), ψ = soil suction, and a, n, m = fitting parameters

Table 4.2. Summary of fitting parameters used to plot the SWCC of Unimin 7030 sand

Parameter a m n

Value 11.415 54.202 5.1322

Soil suction, (kPa)

0 2 4 6 8 10

Vo

lum

etri

c w

ater

co

nte

nt,

0.0

0.1

0.2

0.3

0.4

0.5

Fredlund and Xing (1994)Tempe cell

(ua - uw)b = 4 kPa

(ua - uw)r = 7.8 kPa

Residual zoneBoundary effect zone Transition zone

43

Volume changes can be neglected in cohesionless soils during drying/wetting cycles (i.e.

void ratio, e = constant); therefore, the unit weight of the sand is a function of volumetric

water content as shown in Eq. (4.2). The variation of the unit weight with respect to

volumetric water content (or matric suction) was accounted for in both approaches

discussed in this chapter.

1

1s

w

G e

e

(4.2)

where Gs = specific gravity, and e = void ratio

Effective cohesion and the internal friction angle of the sand were determined from the

direct shear test (Figure 4.4).

Figure 4.4. Direct shear test results on Unimin 7030 sand (modified after Mohamed & Vanapalli 2006)

Normal stress, n (kPa)

0 50 100 150

Sh

ear

stre

ng

th,

(kP

a)

0

50

100

150

Unimin 7030 sand

c' = 0 kPa; ' = 36.2

44

4.2 Estimating the Critical Height with EREPT

The procedure for estimating the critical height of an unsupported vertical trench using

extended Rankine earth pressure theory is as follows:

(a) Set a desired elevation of the GWT.

(b) Set the increment size (i.e. ΔH) for calculating pressures. ΔH = 1-mm was used

in this study for calculating the AEP distribution.

(c) Establish the matric suction distribution with depth by multiplying the unit

weight of water by the distance from the GWT [(ua - uw) = 0 in the saturated zone].

(d) Calculate the distribution of volumetric water content with the computed values

of matric suction by using Eq. (4.1) and the values in Table 4.2.

(e) Calculate the distribution of the unit weight with depth using Eq. (4.2) and the

values in Table 4.1. Use effective unit weight in the saturated zone.

(f) Establish the overburden pressure distribution with depth by multiplying the unit

weight at a specific depth by the depth increment and adding the stress from the

previous increment. This causes the overburden pressure to compound with depth

while accounting for the variation of the unit weight.

(g) Plot the variation of positive earth pressure (i.e. σvKa) with depth. Use effective

stress in the saturated zone.

(h) Convert volumetric water content to the degree of saturation and plot the

variation of negative earth pressure (i.e. 2 aC K ) with depth.

(i) Plot the net AEP distribution as the difference between (g) and (h).

(j) Attain the regression curve for the AEP distribution.

45

(k) Using a tool such as Wolfram Alpha, integrate the regression curve (Figure 4.5)

from the ground surface to the depth where net AEP equals zero to determine A1

(i.e. resultant force in tension zone). Integrate from the depth where AEP = 0 to

the depth which provides A1 = A2 to find the critical height. This may involve

integrating beyond the depth of the GWT. In this case, the first part of A2 is

determined by integrating up to the depth of the GWT. The remaining area is

found by integrating the function for the saturated zone. The critical height is

taken as the depth from the ground surface to where A1 = A2.

Figure 4.5. Example of regression curve for the net AEP distribution with D = 0.6-m

Depth (m)

0.0 0.2 0.4 0.6 0.8 1.0

Net

act

ive

eart

h p

ress

ure

(kP

a)

-4

-2

0

2

4

6

y = 2.656x + 1.5743R² = 1

y = 42.553x4 - 81.394x3 + 57.779x2 - 5.5957x - 2.2089R² = 1

GW

T

Hcr

A1

A2

D

Hcr

46

Figure 4.6 to Figure 4.9 show the distributions of the positive, negative, and net AEP with

depth for D = 0.3, 0.6, 1.0, and 1.5-m, respectively. The absolute value of negative earth

pressure is plotted such that the magnitude of the positive and negative earth pressures can

be compared more easily. The negative sign in the net AEP distribution indicates the

tension zone, and the positive sign indicates the compression zone. When the lateral earth

pressure is zero, the stress-state is transitioning from tension to compression or vice versa.

When D = 1.0-m, the magnitude of negative earth pressure is not sufficient to match the

positive pressure. Also, in the case with D = 1.5-m, the net AEP is positive throughout the

entire soil profile. This indicates that an unsupported vertical trench fails immediately upon

initiating the excavation (i.e. Hcr = 0-m).

47

Figure 4.6. Positive, negative, and net AEP distribution (D = 0.3-m)

Pressure (kPa)

-3 -2 -1 0 1 2 3 4

Dep

th (

m)

0.0

0.1

0.2

0.3

0.4

0.5

Positive earth pressure (vKa)

Negative earth pressure ( )Net active earth pressure (vKa )

Zt

a- 2C Ka2C K

48


Pressure (kPa)

-4 -2 0 2 4 6

Dep

th (

m)

0.0

0.2

0.4

0.6

0.8

1.0


Negative earth pressure ( )Net active earth pressure (vKa )

Zt

a2C K

a- 2C K

49


Pressure (kPa)

-2 0 2 4 6

Dep

th (

m)

0.0

0.2

0.4

0.6

0.8

1.0

50


Pressure (kPa)

-3 0 3 6 9

Dep

th (

m)

0.0

0.3

0.6

0.9

1.2

1.5

Positive eath pressure (vKa)

Negative earth pressure ( ) Net active earth pressure (vKa )a- 2C K

a2C K

51

4.3 Estimating the Critical Height with BSM

In this section, the critical height of an unsupported vertical trench in sand was estimated

using Bishop’s simplified method with geotechnical modelling software, SLOPE/W.

Bishop (1955) assumed that the tangential interslice forces are equal and opposite (i.e. X1

= X2 in Figure 4.10) but horizontal forces are not. The FOS is calculated using Eq. (4.3).

Figure 4.10. Forces acting on the ith slice in Bishop’s simplified method of slices (Craig 2004)

sec1tan

tan tansin1

ii i i i i i

i i ii ii

FOS c W uW

FOS

(4.3)

where FOS = factor of safety, Wi = weight of ith slice, αi = inclination of ith slice base, and

β = width of ith slice

52

This method only satisfies moment equilibrium because interslice shear forces are ignored,

which implies rotational failure is assumed to happen about a circular slip surface (i.e.

negligible internal shearing occurs). This sort of mechanism was deemed plausible for a

vertical trench being excavated into sand with apparent cohesion. An iterative procedure is

required to solve for equilibrium. The initial guess for the FOS in SLOPE/W is one, in

which case the shear strength, and a new FOS are computed. The program continues

iterating until convergence occurs at a specified tolerance (i.e. 0.001 between values for

the FOS). The critical height was taken as the depth immediately preceding the depth that

showed FOS < 1 in the stability analysis.

In-situ analyses in SIGMA/W (Figure 4.11) are used for setting the initial PWP profile and

the ground stresses. In-situ analyses require Fixed-X displacement boundaries (i.e. hollow

red triangles) on the lateral extents, and Fixed-XY boundaries along the base of the domain.

The stresses calculated in SIGMA/W are not used in calculating the FOS for a Bishop

analysis in SLOPE/W; therefore, meshing and the material model are not important for this

application. However, an in-situ analysis was used as the parent to each SLOPE/W analysis

(Figure 4.12) to establish the PWP profile. This was done for convenience, such that the

GWT only needed to be changed in one analysis file rather than in each individual

excavation stage within SLOPE/W.

53

Figure 4.11. Establishing PWP profile with In-situ analysis in SIGMA/W

Figure 4.12. Analysis tree for Bishop’s simplified method with In-situ as the parent analysis

GWT

-0.46-0.44-0.42-0.40-0.38-0.36-0.34-0.32-0.30-0.28-0.26-0.24-0.22-0.20-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.02

Pore-Water Pressure

-3 - -2 kPa-2 - -1 kPa-1 - 0 kPa0 - 1 kPa1 - 2 kPa

Displacement boundary

(SIGMA/W)

(SLOPE/W)

54

In SIGMA/W, the initial PWP profile can be simulated by defining a GWT and a maximum

negative pressure head based on the assumption that PWP varies hydrostatically with

distance above and below the GWT (Figure 4.13). For instance, if the maximum negative

pressure head is lower than the height of the unsaturated soil layer (i.e. Hmax < Hunsat), the

negative PWP is constant up to ground surface beyond the maximum negative pressure

head. Conversely, if the maximum negative pressure head is greater than the height of the

unsaturated soil layer (i.e. Hmax > Hunsat) the negative PWP increases linearly up to ground

surface. Alternatively, a spatial function can be used to specify the pressure head at discrete

points. The pressure head is determined at intermediate points using linear interpolation.

In this study, an initial GWT was assigned to the problem domain to simulate hydrostatic

PWP distributions above and below the GWT. The maximum negative pressure head was

left as the default setting, 5-m.

u = (y - y )

A

B

C u = (y - y )

Hunsat = Height of unsaturated soil layer

yyw

u = Max. negative pressure

Hmax < Hunsat

Hmax

Hunsat

Hsat

Hmax = Max. negative pressure headHsat = Height of saturated soil layer

Hunsat>

a

b bw w

c cw w

Figure 4.13. Calculation of PWP based on the location of GWT in SIGMA/W

55

For soil with negative PWP (i.e. matric suction), total cohesion is computed with Eq. (4.4)

using effective cohesion and the SWCC. The residual volumetric water content was taken

as 5% of the volumetric water content at saturation for calculating total cohesion in

SLOPE/W. The Mohr-Coulomb failure criterion was used as the material model.

tanra w

s r

C c u u

(4.4)

In SLOPE/W, the shape and positions of trial slip surfaces can be defined by four options:

i) Entry and Exit, ii) Grid and Radius, iii) Block Specified, and iv) Fully Specified. Each

method processes every defined slip surface to find the minimum FOS. The ‘Entry and

Exit’ method was chosen to specify the trial slip surfaces in this chapter (Figure 4.14). The

entry was defined as a range as wide as the excavation depth with an entry point every 10-

mm, and the exit was specified as a point at the toe of the slope. Figure 4.15 to Figure 4.18

show the results for the stability analyses with D = 0.3, 0.6, 1.0, and 1.5-m, respectively.

The contours indicate PWP.

Figure 4.14. Schematic of the ‘Entry and Exit’ slip surface method in SLOPE/W

Radius points

Entry range

Exit point

56

Figure 4.15. Slope stability analysis in SLOPE/W using Bishop’s simplified method (D = 0.3-m)


-2

-1

1.02

-0.30-0.28-0.26-0.24-0.22-0.20-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.02

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

-5

-4

-3

-2

-1

1.01

-0.58-0.54-0.50-0.46-0.42-0.38-0.34-0.30-0.26-0.22-0.18-0.14-0.10-0.06-0.02

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

57



-9.5

-9

0.36

-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01 Factor of Safety

≤ 0.35 - 0.450.45 - 0.550.55 - 0.650.65 - 0.75≥ 0.75

-14.5

-14

0.36

-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01

Factor of Safety

≤ 0.35 - 0.450.45 - 0.550.55 - 0.650.65 - 0.75≥ 0.75

58

4.4 Comparison of Critical Heights from EREPT and BSM

Multiple analyses were conducted with the GWT at various elevations to establish a

relationship between the depth of the GWT and the critical height. Table 4.3 summarizes

the critical heights estimated using extended Rankine earth pressure theory and Bishop’s

simplified method. Based on the results in Table 4.3, the variation of the critical height

with respect to the depth of the GWT in Unimin 7030 sand is plotted in Figure 4.19.

Table 4.3. Variation of the critical height in Unimin 7030 sand

GWT (m)

Critical Height (m)

Extended Rankine Earth Pressure Theory Bishop's Simplified Method

0 0.00 0.00

0.1 0.11 0.12

0.2 0.20 0.23

0.3 0.30 0.35

0.4 0.39 0.46

0.5 0.48 0.57

0.6 0.57 0.65

0.7 0.64 0.70

0.8 0.68 0.71

0.9 0.00 0.04

1 0.00 0.00

1.1 0.00 0.00

1.2 0.00 0.00

1.3 0.00 0.00

1.4 0.00 0.00

1.5 0.00 0.00

59

The results from both methods show good agreement. The error ranged from 4%

(0.03/0.68-m) with D = 0.8-m, to 19% (0.09/0.48-m) with D = 0.5-m. Both methodologies

show that the critical height increases gradually as the depth of the GWT increases up to

0.8-m, and then drops to zero as the depth of the GWT is further increased. The minimum

slip surface depth was set at 0.01-m in SLOPE/W, therefore the FOS could not be generated

for the 0.01-m excavation stage. In which case, the critical height was taken as 0-m when

the 0.02-m excavation stage showed FOS < 1 (e.g. Figure 4.17 and Figure 4.18). This

behaviour can be explained by the variation of total cohesion in the sand with respect to

matric suction. The contribution of matric suction towards total cohesion increases with

increasing matric suction, and then starts decreasing as residual suction is approached (7.8-

kPa for Unimin 7030 sand; Vanapalli et al. 1996). In other words, total cohesion becomes

zero in the sand near the top of the trench when the depth of the GWT is relatively deep (>

0.8-m), and shear strength becomes fully dependant on frictional resistance. This indicates

that trench failure occurs at a shallow depth when the GWT is deep enough to create a

residual zone near the ground surface.

For depths of the GWT up to 0.7-m, the critical heights estimated using extended Rankine

earth pressure theory are slightly greater than the depths of the GWT. This may not be

realistic since the sand used in the present study does not have effective cohesion and an

excavation below the GWT can initiate a localized failure near the toe of the slope (i.e.

slough-in/cave-in, Figure 2.5(c)), which leads to a general failure. In other words, these

methods do not account for localized failures that may trigger general failures.

It is interesting to note that the results show close agreement even though the respective

methods are inherently different. Bishop’s simplified method only satisfies moment

60

equilibrium and assumes a circular slip surface, and the shear strength is computed at the

base of each slice along the slip surface. Extended Rankine earth pressure theory is purely

based on force equilibrium. The lateral pressure along the excavation face is computed and

the sum of the forces in the horizontal direction is set to zero to solve for the critical height.

The results are likely close because failure naturally occurs at a relatively shallow depth

and the soil mass is approximately the same regardless of the assumed failure surface.

Figure 4.19. Variation of the critical height with respect to the depth of the GWT using extended Rankine earth pressure theory and Bishop’s simplified method (Unimin 7030 sand)

Depth of GWT (m)

0.0 0.5 1.0 1.5

Hc

r (m

)

0.0

0.2

0.4

0.6

0.8

1.0

Extended Rankine TheoryBishop's Simplified Method

Critical Height = Depth of GWT

61

4.5 Summary and Conclusions

In this chapter, an attempt is made to estimate the critical height of an unsupported vertical

trench in an unsaturated sand using two methodologies; extended Rankine earth pressure

theory, and Bishop’s simplified method. The results showed that the critical height

increases with increasing depth of the GWT up to a point (0.8-m), and then decreases

significantly as the depth of the GWT is further increased. This is attributed to the fact that

total cohesion of an unsaturated sand becomes close to zero as matric suction exceeds the

residual suction value. The agreement between the results suggests that the critical height

of an unsupported vertical trench in unsaturated sand is strongly influenced by the variation

of shear strength with respect to matric suction.

62

CHAPTER 5

ESTIMATING THE CRITICAL HEIGHT OF UNSUPPORTED TRENCHES WITH DIFFERENT WALL SLOPES IN SAND

In Chapter 4, trench stability was estimated without considering the change in PWP due to

excavating. In practice, excavating a trench relieves stress in the soil within the proximity

of the excavation, which results in elastic rebound. This phenomenon leads to an increase

in void ratio and a temporary decrease in PWP. The shear strength of an unsaturated soil

can either increase or decrease depending on the initial matric suction distribution before

excavating. For example, if a trench is excavated into a soil where matric suction is close

to or within the residual zone, a further increase in matric suction can lead to a decrease in

shear strength and therefore the critical height. Conversely, if most soil above the GWT is

within the boundary effect or transition zone, a temporary decrease in PWP may

temporarily increase the shear strength of the soil. However, the probability of the trench

failing increases with time as negative PWP (i.e. matric suction) dissipates.

Coupled stress-PWP analyses (hereafter referred to as coupled analyses) are conducted in

this chapter to investigate the critical height of trenches considering the influence of PWP

redistribution caused by excavating. Four different wall slopes (1.5V:1H, 2V:1H, 3V:1H,

and 90°) and a vertical trench with the top 0.3-m sloped 1:1 (denoted as 90***, Figure 5.1)

were considered with multiple depths of the GWT (0, 0.3, 0.5, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5,

and 2.0-m) in Unimin 7030 sand (Table 4.1). For comparison, the critical heights were also

estimated using the LEM (Morgenstern-Price method) for the same excavation scenarios

used in the coupled analyses.

63

Figure 5.1. Different types of trenches considered in this chapter

5.1 Estimating the Critical Height with the Finite Element Approach

5.1.1 Hydraulic Conductivity Function

The hydraulic conductivity of a soil is maximized when the soil is saturated. As a soil

desaturates, air gradually replaces the voids initially occupied by water and the tortuosity

of the flow path increases. This phenomenon increases the resistance to flow through the

voids. In other words, the hydraulic conductivity of a soil is a function of the degree of

saturation (or matric suction) and decreases with increasing matric suction. However,

measuring the hydraulic conductivity for various matric suction values is time-consuming

90°

90***

V

HV:H slope

64

and requires elaborate testing equipment. For this reason, the hydraulic conductivity

function proposed by Fredlund et al. (1994), Eq. (5.1), was used in this chapter to estimate

the variation of hydraulic conductivity with respect to soil suction. Figure 5.2 shows the

hydraulic conductivity function of Unimin 7030 sand.

1

i

i

i

i

yNy

yi j

unsat sat yNs y

yi

ee

ek k

ee

e

(5.1)

where

kunsat = the calculated conductivity for a specified water content or matric suction,

ksat = the measured conductivity for saturated condition,

y = a variable of integration representing the logarithm of negative PWP,

i = the interval between the range of j to N,

j = the least negative PWP to be described by the final function,

N = the maximum negative PWP to be described by the final function,

= the suction corresponding to the jth interval,

’ = the first derivative of Eq. (4.1)

65

Figure 5.2. Hydraulic conductivity function of Unimin 7030 sand

5.1.2 Analysis in SIGMA/W

The PWP distribution and the initial stresses (i.e. gravity body loads) first must be

established with an In-situ analysis prior to excavating (Figure 4.11). A hydrostatic PWP

distribution was created by setting a static GWT. The excavations were staged using a

coupled analysis. Deformations and stress distributions resulting from each excavation

stage were calculated using effective stress parameters and the elastic-plastic constitutive

model.

Figure 5.3 shows the boundary conditions required to perform a coupled analysis in

SIGMA/W. The left and right ends are restrained in the X-direction (i.e. hollow red

Matric Suction (kPa)

0.1 1.0 10.0

Hyd

rau

lic

Co

nd

uct

ivit

y, k

(m

/s)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

66

triangles), and the base of the domain is restrained in both the X and Y directions. Total

head boundaries equal to the initial water table elevation were placed along the lateral

extents of the soil region (i.e. solid blue dots). This allows the GWT to fluctuate in response

to excavating while maintaining constant hydraulic head along the extents of the domain.

The mesh is created of 0.1 × 0.25-m elements in the immediate surroundings of the

excavation, and transitions to 1 × 1-m elements along the extents of the domain. A finite

element mesh pattern of ‘quads & triangles’ was used to provide a smooth transition

between areas of interest. The main reason for using different element sizes was to save on

computation time. Mesh sizes were determined based on a mesh-convergence study

conducted with different element lengths (1-m, 0.25-m, 0.1-m, and 0.05-m) and a 0.1-m

thickness along the excavated surface. The results showed that the critical height is not

affected by the mesh size when finer than 0.25-m. 4-point integration was used for the

quadrilateral elements, and 3-point integration was used for the triangular elements. A

linear interpolation model was used for calculating stresses and deformations at the nodes.

The use of secondary nodes was not necessary for this application.

The previous stage in the excavation was used as the parent analysis to the following, such

that the stress changes and deformations caused by the previous excavations were

compounded as the stages progressed. Excavations were simulated by deactivating regions

in 0.1-m increments. Analyses for each slope and GWT were conducted with 10-s and

1,000-s time steps between excavation stages to investigate the variation of PWP caused

by different excavation rates. Soil parameters used in the analyses are listed in Table 4.1.

67

Figure 5.3. Boundary conditions required for coupled analyses in SIGMA/W

5.1.3 Analysis in SLOPE/W

The FOS was calculated at targeted time steps based on the stress-deformation results from

SIGMA/W (i.e. SIGMA-Stress type analyses were conducted in SLOPE/W). An example

of the analysis tree for 10-s time steps is shown in Figure 5.4. Shear strength is computed

with either Eq. (3.4) or Eq. (3.14) depending on the location of the potential slip surface

and GWT. The mobilized and resisting shear forces are computed for each slice in finite

element analyses, therefore the FOS is computed for each slice. The global FOS is found

by summing all forces along the slip surface (Eq. (5.2)).

Total head boundary

Displacement boundary

Excavated area

68

r

m

SFOS

S

(5.2)

where Sr = total available shear resistance, and Sm = total mobilized shear force along the

base of the slice

Figure 5.4. Example of slope stability analysis tree used in the coupled analyses (10-s time steps between excavations)

time step

excavation stage

69

The ‘Entry and Exit’ slip surface method was used for generating potential slip surfaces.

The exit was specified as a point at the toe of the excavation, and the entry was defined as

a range as wide as the excavation depth with a possible entry point every 10-mm along the

ground surface (Figure 4.14). The critical height was defined as the excavation depth that

showed FOS = 1.0 (e.g. Figure 5.5), or the depth prior to the excavation stage that showed

FOS < 1.0 in the stability analysis. Contours in Figure 5.5 indicates PWP with depth.

Figure 5.5. Example of slope stability analysis using SIGMA-Stress method in SLOPE/W (1.5V:1H)

-8 -4

0

4

8

12

16

1.001

70

5.2 Estimating the Critical Height with the Limit Equilibrium Method

Stability analyses were also conducted with the LEM for the same scenarios used in the

coupled analyses to investigate the differences. Among various solutions to the LEM, the

method proposed by Morgenstern & Price (1965; hereafter referred to as M-P method) was

used in this chapter. Setting up the analysis domain in SLOPE/W for the M-P method is

the same as a Bishop analysis as described in Chapter 4 (Figure 4.11, Figure 4.12, and

Figure 4.14). The interslice forces are statically indeterminate with the LEM, therefore

various solutions exist based on the assumptions made to solve for equilibrium. In Bishop’s

simplified method, the interslice shear forces are assumed to be equal and opposite to solve

for equilibrium. However, the M-P method accounts for both interslice normal and shear

forces, assuming that the interslice shear force is a function of a scaling factor, an interslice

force function, and the interslice normal force, as shown in Eq. (5.3).

X f x E (5.3)

where X = interslice shear force per unit length, E = interslice normal force per unit length,

λ = scaling factor, and f(x) = specified interslice force function

The interslice force function describes how the magnitude of X/E varies across the failure

surface. The assumption regarding the interslice force function may lead to convergence

issues depending on the stress distribution and slope geometry (Ching & Fredlund 1983).

The half-sine interslice force function was used in this chapter, because it was chosen as

the default setting in SLOPE/W due to user experience and intuition (GEO-SLOPE 2015).

The half-sine function causes the slices on the ends of the slip surface to have the lowest

71

shear-to-normal stress ratio, and the slices in the middle have the highest (Figure 5.6). Once

f (x) is specified, λ is varied systematically to determine the value at which moment and

force equilibrium have the same FOS. The corresponding λ value is multiplied by the

specified f(x) to get the applied f(x), which provides the assumed shear-to-normal stress

ratio for each slice (Figure 5.6).

Figure 5.6. Example of half-sine function used in M-P analyses (λ = 0.25)

If λ equals zero, the M-P method becomes the same as Bishop’s simplified method (i.e.

internal shear forces are equal and opposite). Bishop’s simplified method is appropriate

when rotational failure is likely because no slippage between slices is required for a soil

mass to rotate along a circular slip surface. However, significant slippage and internal

shearing must occur for a soil mass to translate along a circular slip surface. Translational

failure becomes more likely in a cohesionless soil as the slope angle approaches the internal

friction angle. Therefore, the FOS can be unrealistic using Bishop’s simplified method for

Slice Number0 5 10 15 20 25 30

Inte

rslic

e F

orc

e F

un

ctio

n

0.0

0.2

0.4

0.6

0.8

1.0

Specified

Applied

72

sloped excavations in sand. This indicates that the M-P method is more appropriate to

analyze trench stability for the cases examined in this chapter. However, it should be noted

that the interslice and slip surface forces assumed in the LEM may not represent in-situ

stress conditions. Nevertheless, the global FOS from the M-P method is normally

acceptable and adequately addresses general failure (Krahn 2003).

Figure 5.7 shows the forces acting on a slice within an arbitrary slip surface and defines all

geometric parameters. Lateral pressure due to water in tension cracks (AR), external point

load (D), and seismic loads (kW), were not considered in the analyses and are therefore

omitted from the equations for calculating the FOS. In saturated soil, the FOS with respect

to moment and force equilibrium is calculated using Eq. (5.4) and Eq. (5.5), respectively.

The normal force is computed using Eq. (5.6).

Figure 5.7. Forces acting on a slice within an arbitrary slip surface (GEO-SLOPE 2015)

73

( ( ) tan )

m

c R N u RFOS

Wx Nf

(5.4)

( cos ( ) tan cos )

sinf

c N uFOS

N

(5.5)

sin sin tan( )

sin tancos

R L

c uW X X

FOSN

FOS

(5.6)

Where FOSm = factor of safety for moment equilibrium, FOSf = factor of safety for force

equilibrium, N = slice base normal force per unit length (FOS = FOSm or FOSf)

Extending Eq. (5.4) and (5.5), SLOPE/W computes the FOS in unsaturated soil for

moment and force equilibrium as shown in Eq. (5.7) and (5.8), respectively (Fredlund &

Krahn 1977, Fredlund et al. 1981). The normal force, N, is computed as shown in Eq. (5.9).

tan tan1 tan

tan tan

b b

w a

m

c R N u u R

FOSWx Nf

(5.7)

tan tancos 1 tan cos

tan tan

sin

b b

w a

f

c N u u

FOSN

(5.8)

sin sin (tan tan ) sin tan( )

sin tancos

b ba w

R L

c u uW X X

FOSN

FOS

(5.9)

74

5.3 Comparison of Limit Equilibrium and Finite Element Approaches

The FOS of a trench may be relatively high immediately after excavating but may decrease

over time as the GWT and PWP return to equilibrium. This sort of scenario may occur in

practice if an excavation is made rapidly to a desired depth and left open for some period.

Figure 5.8 shows the variation of deformation, PWP, and FOS with time for the case of a

1.5V:1H sloped excavation staged in 0.1-m increments up to 1.3-m at 10-s time steps, with

the GWT initially set at 0.7-m. The time step for the 1.3-m stage was increased to 1,000-s

(Figure 5.4) to allow the GWT to rebound. The black arrows represent hydraulic velocity

vectors, and the magnitude decreases with time as PWP approaches equilibrium. The

sequence in Figure 5.8 shows how an excavation may appear stable but fail shortly

thereafter (i.e. 750-s in this example). As discussed previously, removing soil from the

ground relieves confining pressures which results in expansion of the soil adjacent to the

excavated surface. Figure 5.8 clearly shows that the deformations along the excavation face

gradually increase over time. This phenomenon contributes to the decrease in FOS with

time as shown in Figure 5.9.

75

10

-6

2

-2

2

6

14

Dep

th (

m)

-3.0-2.8-2.6-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0

4

8

12

16

-4

Dep

th (

m)

-3.0-2.8-2.6-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0

-6

2

6

10

14

18

-2

De

pth

(m

)

-3.0-2.8-2.6-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0

4

8

12

16

1.30

Dep

th (

m)

-3.0

-2.7

-2.4

-2.1

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

4

8

12

16

20

1.12

Dep

th (

m)

-3.0

-2.7

-2.4

-2.1

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

4

8

12

16

1.32

Dep

th (

m)

-3.0

-2.7

-2.4

-2.1

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

(c) 500s

(b) 250s

(a) 10s

76

Figure 5.8. Variation of deformation, PWP, and FOS with time for (a) 10-s, (b) 250-s, (c) 500-s, and (d) 750-s after 1.3-m excavation stage with initial D = 0.7-m

Figure 5.9. FOS vs. Time for 1.3-m excavation stage (1.5V:1H) with initial D = 0.7-m

Time (s)

0 200 400 600 800 1000 1200

FO

S

0.9

1.0

1.1

1.2

1.3

1.4

1.5

excavation

-6

2

6

10

14

18

-2

Dep

th (

m)

-3.0-2.8-2.6-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0

4

8

12

16

20

1.00

De

pth

(m

)

-3.0

-2.7

-2.4

-2.1

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

(d)

77

Negligible changes in matric suction are observed near the ground surface throughout the

time steps used in the analysis. This implies that the slope failure is governed by the change

in matric suction distribution between the initial GWT and the toe of the slope. The FOS

remains relatively constant during the first 250-s because the length of slip surface below

the GWT is relatively unchanged, even though the GWT increases with time. However,

the FOS drops rapidly with time once the length of slip surface below the GWT exceeds

what can be considered the critical length of the slip surface under saturated condition.

In geotechnical engineering practice, trenches may be excavated at a relatively fast rate

between stages without allowing the GWT to rebound to its original level. However, once

a trench is excavated up to the targeted depth, PWP will gradually approach equilibrium.

This may occur while field workers are completing their jobs in the trench. As shown in

Figure 5.9, a trench may appear stable initially but can fail over time as the GWT rebounds.

It is interesting to note that failure can occur even before PWP reaches equilibrium. Figure

5.8(d) shows FOS = 1 after 750-s without equilibrium being reached, as evidenced by the

hydraulic velocity vectors.

To further investigate how the redistribution of PWP caused by excavating affects the

critical height of unsupported trenches, slope stability analyses were carried out for two

different time steps; namely, 10-s and 1000-s. These time steps were chosen to simulate

fast and slow excavation rates, respectively. Figure 5.10 and Figure 5.11 show the variation

of the critical height with respect to the depth of the GWT for five different excavation

scenarios with 10-s and 1,000-s time steps, respectively. Figure 5.12 shows the variation

of the critical heights estimated using the M-P method for the same scenarios used in the

coupled analyses. The critical height increases as trench slope is reduced regardless of the

78

level of the GWT. This indicates that sloping trenches is an effective way to increase the

critical height. For 10-s time steps, the critical height of the 1.5V:1H slope was estimated

to be 2.6 times higher compared to the vertical trench for some cases. The critical height

of the vertical trench increases as the depth of the GWT increases and then declines sharply

once the GWT is deeper than 1-m.

The critical heights estimated from the three methods are plotted for the 90°, 3V:1H, 2V:1H,

and 1.5V:1H excavation scenarios in Figure 5.13 to Figure 5.16, respectively (90*** is

explained later).

Figure 5.10. Variation of the critical height for different sloping angles in Unimin 7030 sand (10-s time steps, coupled analysis)

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.59090*** 3V:1H2V:1H1.5V:1H

79

Figure 5.11. Variation of the critical height for different sloping angles in Unimin 7030 sand (1,000-s time steps, coupled analysis)

Figure 5.12. Variation of the critical height for different sloping angles in Unimin 7030 sand (M-P method)

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.59090*** 3V:1H2V:1H1.5V:1H

GWT (m)0.0 0.5 1.0 1.5 2.0

Hcr

(m

)

0.0

0.5

1.0

1.5

2.0

2.5

9090*** 3V:1H2V:1H1.5V:1H

80

Figure 5.13. Variation of the critical height for 90° slope in Unimin 7030 sand

Figure 5.14. Variation of the critical height for 3V:1H slope in Unimin 7030 sand

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.510s1,000sM-P

GWT (m)0.0 0.5 1.0 1.5 2.0

Hcr

(m

)

0.0

0.5

1.0

1.5

2.0

2.510s1,000sM-P

81

Figure 5.15. Variation of the critical height for 2V:1H slope in Unimin 7030 sand

Figure 5.16. Variation of the critical height for 1.5V:1H slope in Unimin 7030 sand

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.510s1,000sM-P

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.510s1,000sM-P

82

As expected, the critical heights estimated using the M-P method showed the lowest values.

The critical height decreases as the period between excavation stages increases (i.e. 10-s to

1,000-s time steps). This is because 10-s time step excavations force the GWT to continue

dropping without giving enough time to rebound a significant amount.

When a soil mass is removed from the ground, the soil within the proximity of the

excavation experiences stress relief ranging from K0z to z in the horizontal and vertical

directions, respectively (where z is the vertical stress at a depth z, and K0 is the coefficient

of earth pressure at-rest). This indicates that the stress relief in the soil near the ground

surface is minimal (or close to zero), and PWP changes are negligible. Therefore, as the

slope of a trench decreases, more stress relief occurs along the face of the slopes and the

change in PWP increases accordingly. For this reason, the largest difference in the critical

heights between the coupled analysis (10-s) and the M-P method is observed for the

1.5V:1H slope. For coupled analyses with 1,000-s time steps, most negative excess PWP

dissipates and the critical heights show small discrepancies with the M-P method when the

GWT is less than 1-m. When the GWT is deeper than 1-m, the critical heights estimated

with the M-P method are zero, but the coupled analyses show minimum values of 0.3-m.

Stability analyses based on limit equilibrium conditions are normally more conservative

compared with that of finite element stress-based analyses. However, there are advantages

to using finite element stress-based stability analyses such as i) displacement compatibility

is satisfied and ii) the ground stresses are much closer to reality (Krahn 2003). The main

reason for this difference is related to the concentration of normal and shear stress in the

toe area of a slope as shown in Figure 5.17 and Figure 5.18, respectively. This phenomenon

83

may increase frictional resistance and in turn increase the local FOS in the slices near the

toe of the slope. For this reason, a sloped trench can be can be excavated past the residual

zone near the ground surface even when the GWT is relatively deep (i.e. deeper than 1-m).

The critical height increases for the vertical excavation as the depth of the GWT increases

up to 0.8-m, and then declines sharply thereafter. As explained previously, a deep GWT

creates a residual suction zone near the ground surface which results in a complete loss of

total cohesion (Figure 5.19). To further investigate this behavior, additional analyses were

conducted for a vertical trench with the first 0.3-m of excavation sloped 1:1 (denoted as

90***) as shown in Figure 5.20. The results from the three methods for the 90*** case are

plotted in Figure 5.21. For the M-P method, the maximum attainable critical height in a

vertical trench was increased by 0.1-m and was sustained for an additional 0.1-m increase

in the depth of the GWT by sloping the soil in the residual zone. For the coupled analyses,

the 90*** case provides higher critical heights with the GWT between 0.5 and 1.5-m. This

indicates that benching or sloping the soil in the residual zone can effectively increase the

critical height. Furthermore, sloping the residual zone may be more effective in some cases

than sloping the entire trench wall (Figure 5.10, Figure 5.11, and Figure 5.12).

84

Figure 5.17. Distribution of total normal stress during 0.3-m excavation stage for 1.5V:1H slope with initial D = 1.2-m (10-s time steps)

Figure 5.18. Distribution of shear stress during 0.3-m excavation stage for 1.5V:1H slope with initial D = 1.2-m (10-s time steps)

Dep

th (

m)

-1.2

-0.9

-0.6

-0.3

0.0

Max. Total Stress

≤ 0 - 2 kPa2 - 4 kPa4 - 6 kPa6 - 8 kPa8 - 10 kPa10 - 12 kPa12 - 14 kPa14 - 16 kPa16 - 18 kPa18 - 20 kPa≥ 20 kPa

Stress concentration

Dep

th (

m)

-1.2

-0.9

-0.6

-0.3

0.0

Max. Shear Stress

≤ 0 - 0.5 kPa0.5 - 1 kPa1 - 1.5 kPa1.5 - 2 kPa2 - 2.5 kPa2.5 - 3 kPa3 - 3.5 kPa3.5 - 4 kPa4 - 4.5 kPa4.5 - 5 kPa≥ 5 kPa

Stress concentration

85

Figure 5.19. Vertical trench with D = 0.9-m (M-P method)

Figure 5.20. Partially sloped (1:1) vertical trench with D = 0.9-m (M-P method)

-6

-4

-2

0.55

Dep

th (

m)

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

-0.0Factor of Safety

0.55 - 0.650.65 - 0.750.75 - 0.850.85 - 0.950.95 - 1.051.05 - 1.151.15 - 1.251.25 - 1.351.35 - 1.45≥ 1.45

-2

-6

2

6

0.90

Dep

th (

m)

-0.9

-0.7

-0.5

-0.3

-0.1Factor of Safety

0.90 - 1.001.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.60≥ 1.60

86

Figure 5.21. Variation of the critical height for 90*** slope in Unimin 7030 sand


M-P and coupled analysis methods were used to investigate the influence of various slopes

and matric suction distributions on the critical height of unsupported trenches in an

unsaturated sand. The coupled analyses show that a trench may appear stable initially and

result in failure as PWP returns to equilibrium conditions. Thus, it is more conservative to

assume a hydrostatic PWP distribution when estimating the critical height. Overall, sloping

the trench greatly increases the critical height. This lends credibility to Canadian provincial

standards (Ontario 2017), as they state a 1:1 slope is sufficient for most excavations where

a vertical trench cannot be excavated to a significant depth. The results also show that the

critical height can be almost doubled by flattening the slopes compared to a vertical

excavation. The main advantage of flattening the slope is that a greater critical height can

GWT (m)0.0 0.5 1.0 1.5 2.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.510s1,000sM-P

87

be achieved even when residual suction is reached near the ground surface. The critical

height was increased as the slope angle was reduced; however, it may not always be

possible to provide a gentle slope to the excavation in geotechnical engineering practice

due to limited space. In this case, a combination of sloping and benching can be used

effectively to achieve targeted critical heights, especially when the soil surface is within

the range of residual suction.

88

CHAPTER 6

ESTIMATING THE CRITICAL HEIGHT OF UNSUPPORTED VERTICAL TRENCHES SUBJECTED TO SURCHARGE PRESSURE

In this chapter, the critical height of unsupported vertical trenches in an unsaturated fine-

grained soil is investigated considering practical scenarios. The trenches were assumed to

be excavated into Indian Head till (hereafter referred to as IHT).

In well-developed cities, there are circumstances where trenches must be excavated

adjacent to an existing superstructure to install or remediate municipal infrastructure such

as storm drains or utility lines. In these cases, the critical height of unsupported vertical

trenches is affected by the stress beneath an existing shallow foundation (hereafter referred

to as foundation stress). Hence, analyses were conducted for multiple foundation stresses

(10, 20, 30, and 50-kPa) located at various distances from the edge of an excavation (1, 2,

3, and 5-m) to simulate this practical scenario. Each scenario was modelled with the GWT

at a depth of 0, 1, 2, 3, and 5-m below the shallow foundation (referred to as depth, D).

6.1 Soil Properties

Basic soil properties of IHT are shown in Table 6.1. The grain size distribution curve and

the SWCC of IHT are shown in Figure 6.1 and Figure 6.2, respectively. The SWCC was

best-fitted using Eq. (4.1) and the parameters, a, m, n are shown in the figure. The shear

strength parameters, c’ and ’ determined by Vanapalli et al. (1997) and Oh &Vanapalli

(2010) were used in the analyses.

89

Table 6.1. Basic soil properties of IHT (Vanapalli et al. 1997, Oh & Vanapalli 2010)

Properties Value

Plasticity Index, Ip (%) 15.5

Saturated unit weight, sat (kN/m3) 20.7

Void ratio, e 0.55

Specific gravity, Gs 2.72

Effective cohesion, c’ (kPa) 5

Effective internal friction angle, ’ (˚) 23.1

Figure 6.1. Grain size distribution curve of IHT (Oh &Vanapalli 2010)

Particle size (mm)

0.001 0.01 0.1 1 10

Per

cen

t fi

ner

(%

)

0

10

20

30

40

50

60

70

80

90

100

90

Figure 6.2. SWCC of IHT (Oh & Vanapalli 2018)

6.2 Estimating the Critical Height without Foundation Stress

In this section, critical heights are estimated using extended Rankine earth pressure theory

and Bishop’s simplified method (see sections 4.2 and Error! Reference source not found.

for more details) without considering foundation stress. This scenario represents the case

where an arbitrary foundation stress is sufficiently far away from the trench such that the

critical height of an unsupported vertical trench is not affected by the foundation.

6.2.1 Estimating the Critical Height with EREPT

The positive, negative, and net AEP distributions for cases with D = 0, 1, 3, and 5-m are

shown in Figure 6.3 to Figure 6.6, respectively.

Suction (kPa)

0.1 1 10 100 1000

Deg

ree

of

satu

rati

on

0.0

0.2

0.4

0.6

0.8

1.0

Fredlund and Xing (1994)Measured (Pressure plate)

1

ln

11.687

0.58066

1.4073

m

e nS

e a

a

m

n

91

Figure 6.3. Positive, negative, and net AEP distribution (D = 0-m)

Pressure (kPa)

-10 -5 0 5 10 15

Dep

th (

m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0


Negative earth pressure (2C Ka)

Net earth pressure (vKa - 2C Ka)

zt

92


Pressure (kPa)

-12 -8 -4 0 4 8 12 16

Dep

th (

m)

0.0

0.5

1.0

1.5

2.0

zt




93


Pressure (kPa)

-20 -10 0 10 20 30

Dep

th (

m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

zt




94


Pressure (kPa)

-20 -10 0 10 20 30 40 50

Dep

th (

m)

0

1

2

3

4

5




zt

95

6.2.2 Estimating the Critical Height with BSM

Figure 6.7 to Figure 6.10 show the slope stability analyses using Bishop’s simplified

method with D = 0, 1, 3, and 5-m, respectively.

Figure 6.7. Slope stability analysis in SLOPE/W using Bishop’s simplified method (D = 0-m)


5

10

15

20

25

1.01 2

.6 m

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

0 0 0

5 5

10

-5

1.00

1.8

m

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

96



-25

-20

-15

-10

-5

1.04

2.5

m

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

-45

-40

-35

-30

-25

-20

-15

1.01

3.3

m

Factor of Safety

≤ 1.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.801.80 - 1.90≥ 1.90

97

6.2.3 Comparison of Critical Heights from EREPT and BSM

The critical heights obtained from extended Rankine earth pressure theory and Bishop’s

simplified method are summarized in Table 6.2 for various levels of the GWT. The results

are plotted in Figure 6.11 for better comparison.

Table 6.2. Variation of the critical height in IHT

Depth of GWT (m)

Critical Height (m)

Extended Rankine Earth Pressure Theory Bishop's Simplified Method

0 2.6 2.8

0.3 2.2 2.3

0.5 2.0 2.1

0.7 1.9 2.0

1 1.8 1.9

1.2 1.8 2.0

1.5 1.9 2.0

2 2.0 2.2

2.5 2.3 2.4

3 2.5 2.6

4 3.0 2.9

5 3.3 3.1

98

Figure 6.11. Variation of the critical height with respect to the depth of the GWT using extended Rankine earth pressure theory and Bishop’s simplified method (IHT)

The results show that there is good agreement between the critical heights obtained from

extended Rankine earth pressure theory and Bishop’s simplified method. This suggests that

extended Rankine earth pressure theory is a simple and efficient way to estimate the critical

height of unsupported vertical trenches in both coarse and fine-grained unsaturated soils.

It is interesting to note that the critical height does not continuously increase with

increasing depth of the GWT, but instead decreases gradually to a minimum value when D

= 1-m, and then starts increasing.

Depth of GWT (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Extended Rankine Theory

Bishop's Simplified Method

Critical Height = Depth of GWT

99

In a cohesive soil, net AEP is the sum of positive and negative earth pressure that are caused

by effective weight and total cohesion of the soil, respectively. The SWCC of a cohesive

soil is distributed over a large range of suction and the rate of change in the total cohesion

to that of soil suction is less compared to sandy soils. Hence, when the GWT is at a shallow

depth (i.e. less than 1-m), the contribution of total cohesion towards negative earth pressure

is less than that of effective weight of soil towards positive earth pressure. This becomes

opposite as the GWT is further decreased, which leads to an increase in the critical height.

In the case where the GWT is at the surface, total cohesion is minimized (= effective

cohesion); however, the unit weight of soil is minimized at the same time (i.e. effective

unit weight). For this reason, the critical height with D = 0-m is greater than that with the

GWT at a depth less than 3.5-m in this soil.

6.3 Estimating the Critical Height with Foundation Stress

The variation of the critical height of an unsupported vertical trench subjected to foundation

stress was estimated with the M-P method (details in section 5.2) to simulate excavations

nearby existing foundations. It was assumed that all soil was removed up to the wall of the

structure before proceeding to excavate below the foundation depth regardless of the

foundation stress. The scenario was modelled as shown in Figure 6.12.

100

Figure 6.12. SLOPE/W model used to consider foundation stress

The critical heights were estimated considering three variables; i) level of the GWT (0, 1,

2, 3 and 5-m), ii) foundation stress (10, 20, 30 and 50 kPa), and iii) distance between the

excavation and the foundation stress (0, 1, 2, and 3-m). The entry range of potential slip

surfaces was defined as double the depth of excavation, spanning from the edge of

excavation towards the foundation with an entry point every 10-cm. The exit was specified

as a point at the toe of the slope. An example of a stability analysis is shown in Figure 6.13.

Figure 6.14 to Figure 6.18 show the variation of the critical height with respect to the

distance from the edge of excavation to the foundation for different GWTs.

Distance (m)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Dep

th (

m)

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Foundation stress

Distance between trench and foundation

101

Figure 6.13. Stability analysis considering 50-kPa surcharge at a distance of 3-m (D = 5-m)


0.98

Distance (m)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Dep

th (

m)

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Factor of Safety

≤ 0.90 - 1.001.00 - 1.101.10 - 1.201.20 - 1.301.30 - 1.401.40 - 1.501.50 - 1.601.60 - 1.701.70 - 1.80≥ 1.80

Distance (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

10 kPa20 kPa30 kPa50 kPa

102



Distance (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0


Distance (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5


103



Distance (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0


Distance (m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Hc

r (m

)

0

1

2

3

4


104

The estimated critical heights decrease with increasing foundation stress. The distance at

which the curves converge in Figure 6.14 to Figure 6.18 shall be referred to as the

‘significant distance’. This is the distance at which the critical height is not affected by the

foundation stress. In other words, failure occurs due to the self-weight of the soil

overcoming the resistance from total cohesion, rather than being influenced by the

foundation stress when the significant distance is reached. For this reason, the critical

heights obtained at the significant distance show good agreement (maximum 8%

discrepancy) compared to those found without any applied foundation stress (Figure 6.11).

This discrepancy exists because different LEMs were used to estimate the critical heights

in section 6.2 and 6.3 (i.e. Bishop’s simplified method and M-P method). The significant

distance is approximately 3-m, 2-m, and 3-m, when D = 0-m, 1-m, and 5-m, respectively.

The shallowest critical height occurs when D = 1-m without any applied foundation stress,

therefore the significant distance is also lowest when D = 1-m. This is because the

significant distance is a function of the desired excavation depth in addition to applied

surcharge pressure (i.e. foundation stress).

The critical height vs. depth of the GWT is plotted in Figure 6.19 to Figure 6.22 for

different foundation stresses that are 0, 1, 2, and 3-m away from the excavation,

respectively. Except for when the distance is 0-m, the critical height is at a minimum for D

= 1-m, which is consistent with the case of no foundation stress. Figure 6.22 clearly shows

that the significant distance is no greater than 3-m in IHT for the foundation stresses and

levels of the GWT used in this chapter, as indicated by the overlapping surcharge curves.

105

Figure 6.19. Critical height vs. depth of the GWT with distance of foundation stress from excavation = 0-m


Depth of GWT (m)

0 1 2 3 4 5

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0


Depth of GWT (m)

0 1 2 3 4 5

Hcr

(m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5


106



Depth of GWT (m)

0 1 2 3 4 5

Hc

r (m

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5


Depth of GWT (m)

0 1 2 3 4 5

Hc

r (m

)

0

1

2

3

4


107


The variation of the critical height of an unsupported vertical trench was investigated with

respect to the proximity and magnitude of adjacent foundation stresses. The critical height

is less dependant on the magnitude of the foundation stress as the distance from the edge

of the excavation to the foundation stress is increased. The significant distance was

estimated to be approximately 3-m regardless of the level of the GWT used in this chapter.

The results show that in cases where a trench is excavated near existing superstructures, a

combination of the foundation stress, distance from the excavation to the foundation, and

level of the GWT (i.e. matric suction distribution) should be considered to reliably estimate

the critical height.

108

CHAPTER 7

GENERAL CONCLUSIONS

Being able to assess the critical height helps facilitate more efficient production in trench-

related work. Canadian provinces enforce strict regulations for excavations that workers

are required to enter (e.g. Regulation 181(1) of the Occupational Health & Safety Act

(1991)). In the case where workers are not required to enter the excavation, the regulations

state that contractors can use shoring at their own discretion. For example, machinery alone

may be capable of performing the task in the trench and no workers are subjected to a

collapse; however, a collapse can still damage the equipment and/or disrupt the work. Thus,

knowing the critical height for a jobsite allows the contractor to make informed decisions

with respect to supporting the excavation walls. Although this study provides framework

for estimating the critical height of unsupported trenches, it is recommended to use shoring

if soil deformations in the proximity of a trench (which can damage existing foundations

or utilities) is a concern.

An attempt was made to investigate the influence of the matric suction distribution on the

critical height of unsupported trenches in unsaturated soil. Extended Rankine earth pressure

theory and Bishop’s simplified method were used to independently estimate the variation

of the critical height of an unsupported vertical trench with respect to the depth of the GWT

in Unimin 7030 sand (Chapter 4), and Indian Head till (Chapter 6). The soils have

inherently different shear strength properties (cohesionless and cohesive soils), yet the

critical heights estimated using both methods showed good agreement for both soils. This

109

suggests that extended Rankine earth pressure theory and Bishop’s simplified method are

reliable methods for estimating the critical height of unsupported vertical trenches in

unsaturated soils where it is necessary to extend the effective stress approach.

Coupled stress analyses that account for changes in PWP over time were used to investigate

the critical height of unsupported excavations with various wall slopes and matric suction

distributions in Unimin 7030 sand. The results showed that the FOS is relatively high

immediately after the excavation, and then decreases over time as the PWP approaches

equilibrium conditions. The critical height increases with increasing depth of the GWT up

to a point and then decreases rapidly as the depth of the GWT is further increased,

regardless of the wall slope. This is due to the decrease in the contribution of matric suction

towards shear strength as matric suction of a soil near the ground surface reaches the

residual condition. The critical height was increased as the slope angle was reduced;

however, it may not be always possible to provide a gentle slope to the excavation in

geotechnical engineering practice due to limited work space in the field. In this case, a

combination of sloping and benching can be used effectively to achieve targeted critical

heights, especially when matric suction nearby the soil surface is in the range of residual

suction.

The Morgenstern-Price method was used in SLOPE/W to examine the influence of

foundation stress, the level of the GWT, and the distance between a trench and a foundation.

All soil should be removed up to the wall of the structure before proceeding with an

excavation below the foundation depth. The significant distance (i.e. the distance that the

critical height is not affected by the foundation stress) was estimated to be approximately

110

3-m for the foundation stresses and levels of the GWT used in this study. The significant

distance is a function of excavation depth and the applied surcharge pressure.

Most Canadian provinces impose a maximum allowable depth of 1.2-m for unsupported

vertical trenches, regardless of soil type and field conditions. The results from this study

suggest that imposing a universal safe height is flawed, and the safe height should be

estimated considering soil type, field conditions, and practical scenarios. In Unimin 7030

sand, only a 0.9-m critical height is the best-case scenario for an unsupported vertical

trench, which is less than the safe height recommended by any Canadian province.

Conversely, the worst-case scenario for the critical height in Indian Head till (i.e. 1.8-m) is

greater than the recommended safe height in all Canadian provinces.

In the case where the critical height is much less than what Canadian provinces suggest,

the critical height can be further increased by benching or sloping the trench walls,

especially in the residual zone. Ontario standards (2017) suggest that sloping the walls 1:1

from the base of excavation where it is not possible to excavate vertically to any significant

depth, and the findings from the current study reinforce this suggestion.

In the long-term, the effective cohesion of soil can be lost due to excessive strain, and

apparent cohesion may be lost due to excessive rainfall or evaporation. However, it is

unrealistic to omit these components of strength in the short-term design of unsupported

trenches.

111

7.1 Recommendations for Future Research

The long-term objective of this study is to provide field engineers and contractors with

guidelines for estimating the critical height of unsupported trenches with given field

conditions (i.e. soil type, matric suction distributions) and circumstances (e.g. surcharge

pressure). This research may facilitate more efficient trench related work by assisting

contractors in determining the depth at which shoring or a trench box is required to perform

any necessary tasks.

The findings from this study are based on analytical and numerical analyses, hence it is

necessary to conduct a series of laboratory and/or field tests to validate the findings. There

are also various practical scenarios that should be considered before estimating the critical

height of unsupported trenches for a jobsite by extending the findings from this study. For

example, if a trench must extend deeper than the GWT, de-watering the trench by pumping

induces seepage through the sidewalls of the trench. Seepage forces can decrease the

effective stress at the toe of the slope, and in turn lower shear strength. Another common

scenario that needs to be considered is heavy equipment operating near the top of a slope.

This induces vibration and affects the PWP and matric suction distribution within the soil.

Thus, the influence of seepage forces and dynamic loading should be studied to improve

the reliability of the methodologies for estimating the critical height of an unsupported

vertical trench.

112

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CURRICULUM VITAE

Candidate’s full name: Adin Alexander Richard

Universities attended: B.Sc.E. (Civil), University of New Brunswick, 2016

Journal publications: N/A

Conference presentations: “Critical height of an unsupported vertical trench in an

unsaturated sand”. In Proceedings of the 70th Canadian

Geotechnical Conference. Ottawa, Canada. (2017).

estimating the critical height of unsupported trenches in

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